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Discrete & Continuous Dynamical Systems - A
November 2013 , Volume 33 , Issue 11&12
Special issue
Jerry Goldstein on the occasion of his 70th birthday
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2013, 33(11&12): i-ii
doi: 10.3934/dcds.2013.33.11i
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Abstract:
Jerome Arthur Goldstein (Jerry) was born on August 5, 1941 in Pittsburgh, PA. He attended Carnegie Mellon University, then called Carnegie Institute of Technology, where he earned his Bachelors of Science degree (1963), Masters of Science degree (1964) and Ph.D. (1967) in Mathematics. Jerry took a postdoctoral position at the Institute for Advanced Study in Princeton, followed by an Assistant Professorship at Tulane University. He became a Full Professor at Tulane University in 1975. After twenty-four years there, Jerry moved ``upriver" to join the faculty (and his wife, Gisèle) at Louisiana State University, where he was Professor of Mathematics from 1991 to 1996. In 1996 Jerry and Gisèle moved to the University of Memphis.
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Jerome Arthur Goldstein (Jerry) was born on August 5, 1941 in Pittsburgh, PA. He attended Carnegie Mellon University, then called Carnegie Institute of Technology, where he earned his Bachelors of Science degree (1963), Masters of Science degree (1964) and Ph.D. (1967) in Mathematics. Jerry took a postdoctoral position at the Institute for Advanced Study in Princeton, followed by an Assistant Professorship at Tulane University. He became a Full Professor at Tulane University in 1975. After twenty-four years there, Jerry moved ``upriver" to join the faculty (and his wife, Gisèle) at Louisiana State University, where he was Professor of Mathematics from 1991 to 1996. In 1996 Jerry and Gisèle moved to the University of Memphis.
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2013, 33(11&12): 4795-4810
doi: 10.3934/dcds.2013.33.4795
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We use semigroup techniques to describe the asymptotic behavior of contractive, periodic evolution families on Hilbert spaces. In particular, we show that such evolution families converge almost weakly to a Floquet representation with discrete spectrum.
We use semigroup techniques to describe the asymptotic behavior of contractive, periodic evolution families on Hilbert spaces. In particular, we show that such evolution families converge almost weakly to a Floquet representation with discrete spectrum.
2013, 33(11&12): 4811-4840
doi: 10.3934/dcds.2013.33.4811
+[Abstract](1325)
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The dynamics of the poles of the two--soliton solutions of the modified Korteweg--de Vries equation $$ u_t + 6u^2u_x + u_{xxx} = 0 $$ are investigated. A consequence of this study is the existence of classes of smooth, complex--valued solutions of this equation, defined for $-\infty < x < \infty$, exponentially decreasing to zero as $|x| \to \infty$, that blow up in finite time.
The dynamics of the poles of the two--soliton solutions of the modified Korteweg--de Vries equation $$ u_t + 6u^2u_x + u_{xxx} = 0 $$ are investigated. A consequence of this study is the existence of classes of smooth, complex--valued solutions of this equation, defined for $-\infty < x < \infty$, exponentially decreasing to zero as $|x| \to \infty$, that blow up in finite time.
2013, 33(11&12): 4841-4873
doi: 10.3934/dcds.2013.33.4841
+[Abstract](1496)
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For a general class of nonlinear, dispersive wave equations, existence of periodic, traveling-wave solutions is studied. These traveling waveforms are the analog of the classical cnoidal-wave solutions of the Korteweg-de Vries equation. They are determined to be stable to perturbation of the same period. Their large wavelength limit is shown to be solitary waves.
For a general class of nonlinear, dispersive wave equations, existence of periodic, traveling-wave solutions is studied. These traveling waveforms are the analog of the classical cnoidal-wave solutions of the Korteweg-de Vries equation. They are determined to be stable to perturbation of the same period. Their large wavelength limit is shown to be solitary waves.
2013, 33(11&12): 4875-4890
doi: 10.3934/dcds.2013.33.4875
+[Abstract](1275)
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We study the asymptotic behavior of solutions to variational inequalities with pointwise constraint on the value and gradient of the functions as the domain becomes unbounded. First, as a model problem, we consider the case when the constraint is only on the value of the functions. Then we consider the more general case of constraint also on the gradient. At the end we consider the case when there is no force term which corresponds to Saint-Venant principle for linear problems.
We study the asymptotic behavior of solutions to variational inequalities with pointwise constraint on the value and gradient of the functions as the domain becomes unbounded. First, as a model problem, we consider the case when the constraint is only on the value of the functions. Then we consider the more general case of constraint also on the gradient. At the end we consider the case when there is no force term which corresponds to Saint-Venant principle for linear problems.
2013, 33(11&12): 4891-4921
doi: 10.3934/dcds.2013.33.4891
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This article investigates Hopf bifurcation for a size-structured population dynamic model that is designed to describe size dispersion among individuals in a given population. This model has a nonlinear and nonlocal boundary condition. We reformulate the problem as an abstract non-densely defined Cauchy problem, and study it in the frame work of integrated semigroup theory. We prove a Hopf bifurcation theorem and we present some numerical simulations to support our analysis.
This article investigates Hopf bifurcation for a size-structured population dynamic model that is designed to describe size dispersion among individuals in a given population. This model has a nonlinear and nonlocal boundary condition. We reformulate the problem as an abstract non-densely defined Cauchy problem, and study it in the frame work of integrated semigroup theory. We prove a Hopf bifurcation theorem and we present some numerical simulations to support our analysis.
2013, 33(11&12): 4923-4944
doi: 10.3934/dcds.2013.33.4923
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In this paper we prove the existence and regularity of positive solutions of the homogeneous Dirichlet problem \begin{equation*} -Δ u=g(x,u) in \Omega, u=0 on ∂ \Omega, \end{equation*} where $g(x,u)$ can be singular as $u\rightarrow0^+$ and $0\le g(x,u)\le\frac{\varphi_0(x)}{u^p}$ or $0\le$ $ g(x,u)$ $\le$ $\varphi_0(x)(1+\frac{1}{u^p})$, with $\varphi_0 \in L^m(\Omega), 1 ≤ m.$ There are no assumptions on the monotonicity of $g(x,\cdot)$ and the existence of super- or sub-solutions.
In this paper we prove the existence and regularity of positive solutions of the homogeneous Dirichlet problem \begin{equation*} -Δ u=g(x,u) in \Omega, u=0 on ∂ \Omega, \end{equation*} where $g(x,u)$ can be singular as $u\rightarrow0^+$ and $0\le g(x,u)\le\frac{\varphi_0(x)}{u^p}$ or $0\le$ $ g(x,u)$ $\le$ $\varphi_0(x)(1+\frac{1}{u^p})$, with $\varphi_0 \in L^m(\Omega), 1 ≤ m.$ There are no assumptions on the monotonicity of $g(x,\cdot)$ and the existence of super- or sub-solutions.
2013, 33(11&12): 4945-4965
doi: 10.3934/dcds.2013.33.4945
+[Abstract](1221)
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A class of ultraparabolic equations with delay, arising from age--structured population diffusion, is analyzed. For such equations well--posedness as well as regularity results with respect to the space variables are proved.
A class of ultraparabolic equations with delay, arising from age--structured population diffusion, is analyzed. For such equations well--posedness as well as regularity results with respect to the space variables are proved.
2013, 33(11&12): 4967-4990
doi: 10.3934/dcds.2013.33.4967
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This paper is devoted to abstract second order complete elliptic differential equations set on $\left[ 0,1\right] $ in non-commutative cases. Existence, uniqueness and maximal regularity of the strict solution are proved. The study is performed in $C^{\theta }\left( \left[ 0,1\right] ;X\right) $.
This paper is devoted to abstract second order complete elliptic differential equations set on $\left[ 0,1\right] $ in non-commutative cases. Existence, uniqueness and maximal regularity of the strict solution are proved. The study is performed in $C^{\theta }\left( \left[ 0,1\right] ;X\right) $.
2013, 33(11&12): 4991-5014
doi: 10.3934/dcds.2013.33.4991
+[Abstract](1196)
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In this work, a biharmonic equation with an impedance (non standard) boundary condition and more general equations are considered. The study is performed in the space $L^{p}(-1,0$ $;$ $X)$, $1 < p < \infty $, where $X$ is a UMD Banach space. The problem is obtained through a formal limiting process on a family of boundary and transmission problems $(P^{\delta})_{\delta > 0}$ set in a domain having a thin layer. The limiting problem models, for instance, the bending of a thin plate with a stiffness on a part of its boundary (see Favini et al. [13]).
We build an explicit representation of the solution, then we study its regularity and give a meaning to the non standard boundary condition.
In this work, a biharmonic equation with an impedance (non standard) boundary condition and more general equations are considered. The study is performed in the space $L^{p}(-1,0$ $;$ $X)$, $1 < p < \infty $, where $X$ is a UMD Banach space. The problem is obtained through a formal limiting process on a family of boundary and transmission problems $(P^{\delta})_{\delta > 0}$ set in a domain having a thin layer. The limiting problem models, for instance, the bending of a thin plate with a stiffness on a part of its boundary (see Favini et al. [13]).
We build an explicit representation of the solution, then we study its regularity and give a meaning to the non standard boundary condition.
2013, 33(11&12): 5015-5047
doi: 10.3934/dcds.2013.33.5015
+[Abstract](1107)
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The paper deals with local well--posedness, global existence and blow--up results for reaction--diffusion equations coupled with nonlinear dynamical boundary conditions. The typical problem studied is \[\begin{cases} u_{t}-\Delta u=|u|^{p-2} u in (0,\infty)\times\Omega,\\ u=0 on [0,\infty) \times \Gamma_{0},\\ \frac{\partial u}{\partial\nu} = -|u_{t}|^{m-2}u_{t} on [0,\infty)\times\Gamma_{1},\\ u(0,x)=u_{0}(x) in \Omega \end{cases}\] where $\Omega$ is a bounded open regular domain of $\mathbb{R}^{n}$ ($n\geq 1$), $\partial\Omega=\Gamma_0\cup\Gamma_1$, $2\le p\le 1+2^*/2$, $m>1$ and $u_0\in H^1(\Omega)$, ${u_0}_{|\Gamma_0}=0$. After showing local well--posedness in the Hadamard sense we give global existence and blow--up results when $\Gamma_0$ has positive surface measure. Moreover we discuss the generalization of the above mentioned results to more general problems where the terms $|u|^{p-2}u$ and $|u_{t}|^{m-2}u_{t}$ are respectively replaced by $f\left(x,u\right)$ and $Q(t,x,u_t)$ under suitable assumptions on them.
The paper deals with local well--posedness, global existence and blow--up results for reaction--diffusion equations coupled with nonlinear dynamical boundary conditions. The typical problem studied is \[\begin{cases} u_{t}-\Delta u=|u|^{p-2} u in (0,\infty)\times\Omega,\\ u=0 on [0,\infty) \times \Gamma_{0},\\ \frac{\partial u}{\partial\nu} = -|u_{t}|^{m-2}u_{t} on [0,\infty)\times\Gamma_{1},\\ u(0,x)=u_{0}(x) in \Omega \end{cases}\] where $\Omega$ is a bounded open regular domain of $\mathbb{R}^{n}$ ($n\geq 1$), $\partial\Omega=\Gamma_0\cup\Gamma_1$, $2\le p\le 1+2^*/2$, $m>1$ and $u_0\in H^1(\Omega)$, ${u_0}_{|\Gamma_0}=0$. After showing local well--posedness in the Hadamard sense we give global existence and blow--up results when $\Gamma_0$ has positive surface measure. Moreover we discuss the generalization of the above mentioned results to more general problems where the terms $|u|^{p-2}u$ and $|u_{t}|^{m-2}u_{t}$ are respectively replaced by $f\left(x,u\right)$ and $Q(t,x,u_t)$ under suitable assumptions on them.
2013, 33(11&12): 5049-5058
doi: 10.3934/dcds.2013.33.5049
+[Abstract](1264)
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In this paper we give sufficient conditions ensuring that the space of test functions $C_c^{\infty}(R^N)$ is a core for the operator $$L_0u=\Delta u-Mx\cdot \nabla u+\frac{\alpha}{|x|^2}u=:Lu+\frac{\alpha}{|x|^2}u,$$ and $L_0$ with domain $W_\mu^{2,p}(R^N)$ generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p_\mu(R^N),\,1 < p < \infty$. Here $M$ is a positive definite $N\times N$ hermitian matrix and $\mu$ is the unique invariant measure for the Ornstein-Uhlenbeck operator $L$. The proofs are based on an $L^p$-weighted Hardy's inequality and perturbation techniques.
In this paper we give sufficient conditions ensuring that the space of test functions $C_c^{\infty}(R^N)$ is a core for the operator $$L_0u=\Delta u-Mx\cdot \nabla u+\frac{\alpha}{|x|^2}u=:Lu+\frac{\alpha}{|x|^2}u,$$ and $L_0$ with domain $W_\mu^{2,p}(R^N)$ generates a quasi-contractive and positivity preserving $C_0$-semigroup in $L^p_\mu(R^N),\,1 < p < \infty$. Here $M$ is a positive definite $N\times N$ hermitian matrix and $\mu$ is the unique invariant measure for the Ornstein-Uhlenbeck operator $L$. The proofs are based on an $L^p$-weighted Hardy's inequality and perturbation techniques.
2013, 33(11&12): 5059-5066
doi: 10.3934/dcds.2013.33.5059
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Some deterministic models for prey and predators are considered, when both may become infected, the infection of the prey being either of the SIS or SIR type. We also study a simplified model for surviving predators.
Some deterministic models for prey and predators are considered, when both may become infected, the infection of the prey being either of the SIS or SIR type. We also study a simplified model for surviving predators.
2013, 33(11&12): 5067-5088
doi: 10.3934/dcds.2013.33.5067
+[Abstract](1153)
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A detailed description of the model Hilbert space $L^2(\mathbb{R}; d\Sigma; K)$, where $K$ represents a complex, separable Hilbert space, and $\Sigma$ denotes a bounded operator-valued measure, is provided. In particular, we show that several alternative approaches to such a construction in the literature are equivalent.
These spaces are of fundamental importance in the context of perturbation theory of self-adjoint extensions of symmetric operators, and the spectral theory of ordinary differential operators with operator-valued coefficients.
A detailed description of the model Hilbert space $L^2(\mathbb{R}; d\Sigma; K)$, where $K$ represents a complex, separable Hilbert space, and $\Sigma$ denotes a bounded operator-valued measure, is provided. In particular, we show that several alternative approaches to such a construction in the literature are equivalent.
These spaces are of fundamental importance in the context of perturbation theory of self-adjoint extensions of symmetric operators, and the spectral theory of ordinary differential operators with operator-valued coefficients.
2013, 33(11&12): 5089-5106
doi: 10.3934/dcds.2013.33.5089
+[Abstract](1239)
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We consider a phase-field model of Caginalp type where the free energy depends on the order parameter $\chi$ in a nonlocal way. Therefore, the resulting system consists of the energy balance equation coupled with a nonlinear and nonlocal ODE for $\chi$. Such system has been analyzed by several authors, in particular when the configuration potential is a smooth double-well function. More recently, the first author has established the existence of a finite-dimensional global attractor in the case of a potential defined on $(-1,1)$ and singular at the endpoints. Here we examine both the case of regular potentials as well as the case of physically more relevant singular potentials (e.g., logarithmic). We prove well-posedness results and the eventual global boundedness of solutions uniformly with respect to the initial data. In addition, we show that the separation property holds in the case of singular potentials. Thanks to these results, we are able to demonstrate the existence of a finite-dimensional global attractor in the present cases as well.
We consider a phase-field model of Caginalp type where the free energy depends on the order parameter $\chi$ in a nonlocal way. Therefore, the resulting system consists of the energy balance equation coupled with a nonlinear and nonlocal ODE for $\chi$. Such system has been analyzed by several authors, in particular when the configuration potential is a smooth double-well function. More recently, the first author has established the existence of a finite-dimensional global attractor in the case of a potential defined on $(-1,1)$ and singular at the endpoints. Here we examine both the case of regular potentials as well as the case of physically more relevant singular potentials (e.g., logarithmic). We prove well-posedness results and the eventual global boundedness of solutions uniformly with respect to the initial data. In addition, we show that the separation property holds in the case of singular potentials. Thanks to these results, we are able to demonstrate the existence of a finite-dimensional global attractor in the present cases as well.
2013, 33(11&12): 5107-5141
doi: 10.3934/dcds.2013.33.5107
+[Abstract](1137)
+[PDF](526.7KB)
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We consider the problem of the reconstruction of the source term in a parabolic Cauchy-Dirichlet system in a cylindrical domain. The supplementary information, necessary to determine the unknown part of the source term together with the solution, is given by the knowledge of an integral of the solution with respect to some of the space variables.
We consider the problem of the reconstruction of the source term in a parabolic Cauchy-Dirichlet system in a cylindrical domain. The supplementary information, necessary to determine the unknown part of the source term together with the solution, is given by the knowledge of an integral of the solution with respect to some of the space variables.
2013, 33(11&12): 5143-5151
doi: 10.3934/dcds.2013.33.5143
+[Abstract](1312)
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Consider the Navier-Stokes equations in the rotational framework either on $\mathbb{R}^3$ or on open sets $\Omega \subset \mathbb{R}^3$ subject to Dirichlet boundary conditions. This paper discusses recent well-posedness and ill-posedness results for both situations.
Consider the Navier-Stokes equations in the rotational framework either on $\mathbb{R}^3$ or on open sets $\Omega \subset \mathbb{R}^3$ subject to Dirichlet boundary conditions. This paper discusses recent well-posedness and ill-posedness results for both situations.
2013, 33(11&12): 5153-5166
doi: 10.3934/dcds.2013.33.5153
+[Abstract](1335)
+[PDF](466.9KB)
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We study radial positive solutions to the singular boundary value problem \begin{equation*} \begin{cases} -\Delta_p u = \lambda K(|x|)\frac{f(u)}{u^\beta} \quad \mbox{in}~ \Omega,\\ ~~~u(x) = 0 \qquad \qquad \qquad \qquad~~\mbox{if}~|x|=r_0,\\ ~~~u(x) \rightarrow 0 \qquad\qquad \qquad \mbox{if}~|x|\rightarrow \infty, \end{cases} \end{equation*} where $\Delta_p u =$ div $(|\nabla u|^{p-2}\nabla u)$, $1 < p < N, N >2, \lambda > 0, 0 \leq \beta <1 ,\Omega= \{ x \in \mathbb{R}^{N} : |x| > r_0 \}$ and $ r_0 >0$. Here $f:[0, \infty)\rightarrow (0, \infty)$ is a continuous nondecreasing function such that $\lim_{u\rightarrow \infty} \frac{f(u)}{u^{\beta+p-1}}= 0$ and $ K \in C( (r_0, \infty),(0, \infty) ) $ is such that $\int_{r_0}^{\infty} r^\mu K(r) dr < \infty, $ for some $\mu > p-1$. We establish the existence of multiple positive solutions for certain range of $\lambda$ when $f$ satisfies certain additional assumptions. A simple model that will satisfy our hypotheses is $f(u)=e^{\frac{\alpha u}{\alpha+u}}$ for $ \alpha \gg 1.$ We also extend our results to classes of systems when the nonlinearities satisfy a combined sublinear condition at infinity. We prove our results by the method of sub-super solutions.
We study radial positive solutions to the singular boundary value problem \begin{equation*} \begin{cases} -\Delta_p u = \lambda K(|x|)\frac{f(u)}{u^\beta} \quad \mbox{in}~ \Omega,\\ ~~~u(x) = 0 \qquad \qquad \qquad \qquad~~\mbox{if}~|x|=r_0,\\ ~~~u(x) \rightarrow 0 \qquad\qquad \qquad \mbox{if}~|x|\rightarrow \infty, \end{cases} \end{equation*} where $\Delta_p u =$ div $(|\nabla u|^{p-2}\nabla u)$, $1 < p < N, N >2, \lambda > 0, 0 \leq \beta <1 ,\Omega= \{ x \in \mathbb{R}^{N} : |x| > r_0 \}$ and $ r_0 >0$. Here $f:[0, \infty)\rightarrow (0, \infty)$ is a continuous nondecreasing function such that $\lim_{u\rightarrow \infty} \frac{f(u)}{u^{\beta+p-1}}= 0$ and $ K \in C( (r_0, \infty),(0, \infty) ) $ is such that $\int_{r_0}^{\infty} r^\mu K(r) dr < \infty, $ for some $\mu > p-1$. We establish the existence of multiple positive solutions for certain range of $\lambda$ when $f$ satisfies certain additional assumptions. A simple model that will satisfy our hypotheses is $f(u)=e^{\frac{\alpha u}{\alpha+u}}$ for $ \alpha \gg 1.$ We also extend our results to classes of systems when the nonlinearities satisfy a combined sublinear condition at infinity. We prove our results by the method of sub-super solutions.
2013, 33(11&12): 5167-5176
doi: 10.3934/dcds.2013.33.5167
+[Abstract](1170)
+[PDF](399.2KB)
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The purpose of this paper is to study the nonexistence of positive solutions of the doubly nonlinear equation \[\begin{cases} \frac{\partial u}{\partial t}=\nabla_{\gamma}\cdot (u^{m-1}|\nabla_{\gamma} u|^{p-2}\nabla_{\gamma} u) +Vu^{m+p-2} & \text{in}\quad \Omega \times (0, T ) ,\\ u(x,0)=u_{0}(x)\geq 0 & \text{in} \quad\Omega, \\ u(x,t)=0 & \text{on}\quad \partial\Omega\times (0, T),\end{cases}\] where $\nabla_{\gamma}=(\nabla_x, |x|^{2\gamma}\nabla_y)$, $x\in \mathbb{R}^d, y\in \mathbb{R}^k$, $\gamma>0$, $\Omega$ is a metric ball in $\mathbb{R}^{N}$, $V\in L_{\text{loc}}^1(\Omega)$, $m\in \mathbb{R}$, $1 < p < d+k$ and $m + p - 2 > 0$. The exponents $q^{*}$ are found and the nonexistence results are proved for $q^{*} ≤ m+p < 3$.
The purpose of this paper is to study the nonexistence of positive solutions of the doubly nonlinear equation \[\begin{cases} \frac{\partial u}{\partial t}=\nabla_{\gamma}\cdot (u^{m-1}|\nabla_{\gamma} u|^{p-2}\nabla_{\gamma} u) +Vu^{m+p-2} & \text{in}\quad \Omega \times (0, T ) ,\\ u(x,0)=u_{0}(x)\geq 0 & \text{in} \quad\Omega, \\ u(x,t)=0 & \text{on}\quad \partial\Omega\times (0, T),\end{cases}\] where $\nabla_{\gamma}=(\nabla_x, |x|^{2\gamma}\nabla_y)$, $x\in \mathbb{R}^d, y\in \mathbb{R}^k$, $\gamma>0$, $\Omega$ is a metric ball in $\mathbb{R}^{N}$, $V\in L_{\text{loc}}^1(\Omega)$, $m\in \mathbb{R}$, $1 < p < d+k$ and $m + p - 2 > 0$. The exponents $q^{*}$ are found and the nonexistence results are proved for $q^{*} ≤ m+p < 3$.
2013, 33(11&12): 5177-5187
doi: 10.3934/dcds.2013.33.5177
+[Abstract](1147)
+[PDF](310.6KB)
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We investigate a class of bivariate coagulation-fragmentation equations. These equations describe the evolution of a system of particles that are characterised not only by a discrete size variable but also by a shape variable which can be either discrete or continuous. Existence and uniqueness of strong solutions to the associated abstract Cauchy problems are established by using the theory of substochastic semigroups of operators.
We investigate a class of bivariate coagulation-fragmentation equations. These equations describe the evolution of a system of particles that are characterised not only by a discrete size variable but also by a shape variable which can be either discrete or continuous. Existence and uniqueness of strong solutions to the associated abstract Cauchy problems are established by using the theory of substochastic semigroups of operators.
2013, 33(11&12): 5189-5202
doi: 10.3934/dcds.2013.33.5189
+[Abstract](1312)
+[PDF](418.0KB)
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We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in $\mathbb{R}^n$. Global Well-posedness of solutions is shown by applying the theory of maximal parabolic regularity of type $L_p$. In addition, we prove exponential decay rates for strong solutions and their derivatives.
We consider a quasilinear PDE system which models nonlinear vibrations of a thermoelastic plate defined on a bounded domain in $\mathbb{R}^n$. Global Well-posedness of solutions is shown by applying the theory of maximal parabolic regularity of type $L_p$. In addition, we prove exponential decay rates for strong solutions and their derivatives.
2013, 33(11&12): 5203-5216
doi: 10.3934/dcds.2013.33.5203
+[Abstract](1446)
+[PDF](440.3KB)
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For a strongly continuous operator semigroup on a Banach space, we revisit a quantitative version of Datko's Theorem and the estimates for the constant $M$ satisfying the inequality $||T(t)|| ≤ M e^{\omega t}$, for all $t\ge0$, in terms of the norm of the convolution and other operators involved in Datko's Theorem. We use techniques recently developed by B. Helffer and J. Sjöstrand for the Hilbert space case to estimate $M$ in terms of the norm of the resolvent of the generator of the semigroup in the right half-plane.
For a strongly continuous operator semigroup on a Banach space, we revisit a quantitative version of Datko's Theorem and the estimates for the constant $M$ satisfying the inequality $||T(t)|| ≤ M e^{\omega t}$, for all $t\ge0$, in terms of the norm of the convolution and other operators involved in Datko's Theorem. We use techniques recently developed by B. Helffer and J. Sjöstrand for the Hilbert space case to estimate $M$ in terms of the norm of the resolvent of the generator of the semigroup in the right half-plane.
2013, 33(11&12): 5217-5252
doi: 10.3934/dcds.2013.33.5217
+[Abstract](1603)
+[PDF](601.1KB)
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We consider a second-order hyperbolic equation defined on an open bounded domain $\Omega$ in $\mathbb{R}^n$ for $n \geq 2$, with $C^2$-boundary $\Gamma = \partial \Omega = \overline{\Gamma_0 \cup \Gamma_1}$, $\Gamma_0 \cap \Gamma_1 = \emptyset$, subject to non-homogeneous Dirichlet boundary conditions for the entire boundary $\Gamma$. We then study the inverse problem of determining both the damping and the potential (source) coefficients simultaneously, in one shot, by means of an additional measurement of the Neumann boundary trace of the solution, in a suitable, explicit sub-portion $\Gamma_1$ of the boundary $\Gamma$, and over a computable time interval $T > 0$. Under sharp conditions on the complementary part $\Gamma_0 = \Gamma \backslash \Gamma_1$, $T > 0$, and under sharp regularity requirements on the data, we establish the two canonical results in inverse problems: (i) uniqueness and (ii) Lipschitz-stability. The latter (ii) is the main result of the paper. Our proof relies on a few main ingredients: (a) sharp Carleman estimates at the $H^1(\Omega) \times L^2(\Omega)$-level for second-order hyperbolic equations [23], originally introduced for control theory issues; (b) a correspondingly implied continuous observability inequality at the same energy level [23]; (c) sharp interior and boundary regularity theory for second-order hyperbolic equations with Dirichlet boundary data [14,15,16]. The proof of the linear uniqueness result (Section 3) also takes advantage of a convenient tactical route ``post-Carleman estimates" proposed by V. Isakov in [8, Thm. 8.2.2, p. 231]. Expressing the final results for the nonlinear inverse problem directly in terms of the data offers an additional challenge.
We consider a second-order hyperbolic equation defined on an open bounded domain $\Omega$ in $\mathbb{R}^n$ for $n \geq 2$, with $C^2$-boundary $\Gamma = \partial \Omega = \overline{\Gamma_0 \cup \Gamma_1}$, $\Gamma_0 \cap \Gamma_1 = \emptyset$, subject to non-homogeneous Dirichlet boundary conditions for the entire boundary $\Gamma$. We then study the inverse problem of determining both the damping and the potential (source) coefficients simultaneously, in one shot, by means of an additional measurement of the Neumann boundary trace of the solution, in a suitable, explicit sub-portion $\Gamma_1$ of the boundary $\Gamma$, and over a computable time interval $T > 0$. Under sharp conditions on the complementary part $\Gamma_0 = \Gamma \backslash \Gamma_1$, $T > 0$, and under sharp regularity requirements on the data, we establish the two canonical results in inverse problems: (i) uniqueness and (ii) Lipschitz-stability. The latter (ii) is the main result of the paper. Our proof relies on a few main ingredients: (a) sharp Carleman estimates at the $H^1(\Omega) \times L^2(\Omega)$-level for second-order hyperbolic equations [23], originally introduced for control theory issues; (b) a correspondingly implied continuous observability inequality at the same energy level [23]; (c) sharp interior and boundary regularity theory for second-order hyperbolic equations with Dirichlet boundary data [14,15,16]. The proof of the linear uniqueness result (Section 3) also takes advantage of a convenient tactical route ``post-Carleman estimates" proposed by V. Isakov in [8, Thm. 8.2.2, p. 231]. Expressing the final results for the nonlinear inverse problem directly in terms of the data offers an additional challenge.
2013, 33(11&12): 5253-5271
doi: 10.3934/dcds.2013.33.5253
+[Abstract](1121)
+[PDF](388.5KB)
Abstract:
We prove the existence of a spatial coefficient in front of a nonlinear term in a one-dimensional wave equation when, in addition to classical initial and boundary condition, an integral mean involving the displacement is prescribed.
We prove the existence of a spatial coefficient in front of a nonlinear term in a one-dimensional wave equation when, in addition to classical initial and boundary condition, an integral mean involving the displacement is prescribed.
2013, 33(11&12): 5273-5292
doi: 10.3934/dcds.2013.33.5273
+[Abstract](1344)
+[PDF](458.3KB)
Abstract:
We consider a linear evolution model describing a piezoelectric phenomenon under thermal effects as suggested by R. Mindlin [13] and W. Nowacki [16]. We prove the equivalence between exponential decay of the total energy and an observability inequality for an anisotropic elastic wave system. Our strategy is to use a decoupling method to reduce the problem to an equivalent observability inequality for an anisotropic elastic wave system and assume a condition which guarantees that the corresponding elliptic operator has no eigenfunctions with null divergence.
We consider a linear evolution model describing a piezoelectric phenomenon under thermal effects as suggested by R. Mindlin [13] and W. Nowacki [16]. We prove the equivalence between exponential decay of the total energy and an observability inequality for an anisotropic elastic wave system. Our strategy is to use a decoupling method to reduce the problem to an equivalent observability inequality for an anisotropic elastic wave system and assume a condition which guarantees that the corresponding elliptic operator has no eigenfunctions with null divergence.
2013, 33(11&12): 5293-5303
doi: 10.3934/dcds.2013.33.5293
+[Abstract](1204)
+[PDF](331.5KB)
Abstract:
For a given autonomous time-dependent system that generates either a global, in time, semigroup or else only a local, in time, semigroup, a test involving a linear eigenvalue problem is given which determines which of `global' or `local' holds. Numerical examples are given. A linear transformation $A$ is defined so that one has `global' or `local' depending on whether $A$ does not or does have a positive eigenvalue. There is a possible application to Navier-Stokes problems..
For a given autonomous time-dependent system that generates either a global, in time, semigroup or else only a local, in time, semigroup, a test involving a linear eigenvalue problem is given which determines which of `global' or `local' holds. Numerical examples are given. A linear transformation $A$ is defined so that one has `global' or `local' depending on whether $A$ does not or does have a positive eigenvalue. There is a possible application to Navier-Stokes problems..
2013, 33(11&12): 5305-5317
doi: 10.3934/dcds.2013.33.5305
+[Abstract](1283)
+[PDF](375.8KB)
Abstract:
We show that for all $q\ge 1$ and $1\le i \le q$ there exist pairwise conjugate complex numbers $b_{q,i}$ and $\lambda_{q,i}$ with $\mbox{Re} (\lambda_{q,i}) > 0$ such that for any generator $(A, D(A))$ of a bounded, strongly continuous semigroup $T(t)$ on Banach space $X$ with resolvent $R(\lambda,A) := (\lambda I-A)^{-1}$ the expression $\frac{b_{q,1}}{t}R(\frac{\lambda_{q,1}}{t},A) + \frac{b_{q,2}}{t}R(\frac{\lambda_{q,2}}{t},A) + \cdots + \frac{b_{q,q}}{t}R(\frac{\lambda_{q,q}}{t},A)$ provides an excellent approximation of the semigroup $T(t)$ on $D(A^{2q-1})$. Precise error estimates as well as applications to the numerical inversion of the Laplace transform are given.
We show that for all $q\ge 1$ and $1\le i \le q$ there exist pairwise conjugate complex numbers $b_{q,i}$ and $\lambda_{q,i}$ with $\mbox{Re} (\lambda_{q,i}) > 0$ such that for any generator $(A, D(A))$ of a bounded, strongly continuous semigroup $T(t)$ on Banach space $X$ with resolvent $R(\lambda,A) := (\lambda I-A)^{-1}$ the expression $\frac{b_{q,1}}{t}R(\frac{\lambda_{q,1}}{t},A) + \frac{b_{q,2}}{t}R(\frac{\lambda_{q,2}}{t},A) + \cdots + \frac{b_{q,q}}{t}R(\frac{\lambda_{q,q}}{t},A)$ provides an excellent approximation of the semigroup $T(t)$ on $D(A^{2q-1})$. Precise error estimates as well as applications to the numerical inversion of the Laplace transform are given.
2013, 33(11&12): 5319-5325
doi: 10.3934/dcds.2013.33.5319
+[Abstract](1258)
+[PDF](258.5KB)
Abstract:
The Thomas-Fermi equation arises from the earliest density functional approximation for the ground-state energy of a many-electron system. Its solutions have been carefully studied by mathematicians, including J.A. Goldstein. Here we will review the approximation and its validity conditions from a physics perspective, explaining why the theory correctly describes the core electrons of an atom but fails to bind atoms to form molecules and solids. The valence electrons are poorly described in Thomas-Fermi theory, for two reasons: (1) This theory neglects the exchange-correlation energy, ``nature's glue". (2) It also makes a local density approximation for the kinetic energy, which neglects important shell-structure effects in the exact kinetic energy that are responsible for the structure of the periodic table of the elements. Finally, we present a tentative explanation for the fact that the shell-structure effects are relatively unimportant for the exact exchange energy, which can thus be more usefully described by a local density or semilocal approximation (as in the popular Kohn-Sham theory): The exact exchange energy from the occupied Kohn-Sham orbitals has an extra sum over orbital labels and an extra integration over space, in comparison to the kinetic energy, and thus averages out more of the atomic individuality of the orbital oscillations.
The Thomas-Fermi equation arises from the earliest density functional approximation for the ground-state energy of a many-electron system. Its solutions have been carefully studied by mathematicians, including J.A. Goldstein. Here we will review the approximation and its validity conditions from a physics perspective, explaining why the theory correctly describes the core electrons of an atom but fails to bind atoms to form molecules and solids. The valence electrons are poorly described in Thomas-Fermi theory, for two reasons: (1) This theory neglects the exchange-correlation energy, ``nature's glue". (2) It also makes a local density approximation for the kinetic energy, which neglects important shell-structure effects in the exact kinetic energy that are responsible for the structure of the periodic table of the elements. Finally, we present a tentative explanation for the fact that the shell-structure effects are relatively unimportant for the exact exchange energy, which can thus be more usefully described by a local density or semilocal approximation (as in the popular Kohn-Sham theory): The exact exchange energy from the occupied Kohn-Sham orbitals has an extra sum over orbital labels and an extra integration over space, in comparison to the kinetic energy, and thus averages out more of the atomic individuality of the orbital oscillations.
2013, 33(11&12): 5327-5345
doi: 10.3934/dcds.2013.33.5327
+[Abstract](1766)
+[PDF](891.0KB)
Abstract:
The aim of this article is to study a model of two superposed layers of fluid governed by the shallow water equations in space dimension one. Under some suitable hypotheses the governing equations are hyperbolic. We introduce suitable boundary conditions and establish a result of existence and uniqueness of smooth solutions for a limited time for this model.
The aim of this article is to study a model of two superposed layers of fluid governed by the shallow water equations in space dimension one. Under some suitable hypotheses the governing equations are hyperbolic. We introduce suitable boundary conditions and establish a result of existence and uniqueness of smooth solutions for a limited time for this model.
2013, 33(11&12): 5347-5377
doi: 10.3934/dcds.2013.33.5347
+[Abstract](1185)
+[PDF](480.6KB)
Abstract:
The main goal of this paper is to prove existence of global solutions in time for an Allen-Cahn-Gurtin model of pseudo-parabolic type. Local solutions were known to ``blow up" in some sense in finite time. It is proved that the equation is actually governed by a monotone-like operator. It turns out to be multivalued and measure-valued. The measures are singular with respect to the Lebesgue measure. This operator allows to extend the local solutions globally in time and to fully solve the evolution problem. The asymptotic behavior is also analyzed.
The main goal of this paper is to prove existence of global solutions in time for an Allen-Cahn-Gurtin model of pseudo-parabolic type. Local solutions were known to ``blow up" in some sense in finite time. It is proved that the equation is actually governed by a monotone-like operator. It turns out to be multivalued and measure-valued. The measures are singular with respect to the Lebesgue measure. This operator allows to extend the local solutions globally in time and to fully solve the evolution problem. The asymptotic behavior is also analyzed.
2013, 33(11&12): 5379-5405
doi: 10.3934/dcds.2013.33.5379
+[Abstract](1230)
+[PDF](515.3KB)
Abstract:
We prove strong convergence to singular limits for a linearized fully inhomogeneous Stefan problem subject to surface tension and kinetic undercooling effects. Different combinations of $\sigma \to \sigma_0$ and $\delta\to\delta_0$, where $\sigma,\sigma_0\ge 0$ and $\delta,\delta_0\ge 0$ denote surface tension and kinetic undercooling coefficients respectively, altogether lead to five different types of singular limits. Their strong convergence is based on uniform maximal regularity estimates.
We prove strong convergence to singular limits for a linearized fully inhomogeneous Stefan problem subject to surface tension and kinetic undercooling effects. Different combinations of $\sigma \to \sigma_0$ and $\delta\to\delta_0$, where $\sigma,\sigma_0\ge 0$ and $\delta,\delta_0\ge 0$ denote surface tension and kinetic undercooling coefficients respectively, altogether lead to five different types of singular limits. Their strong convergence is based on uniform maximal regularity estimates.
2013, 33(11&12): 5407-5428
doi: 10.3934/dcds.2013.33.5407
+[Abstract](1124)
+[PDF](406.5KB)
Abstract:
Several results from differential geometry of hypersurfaces in $\mathbb{R}^n$ are derived to form a tool box for the direct mapping method. The latter technique has been widely employed to solve problems with moving interfaces, and to study the asymptotics of the induced semiflows.
Several results from differential geometry of hypersurfaces in $\mathbb{R}^n$ are derived to form a tool box for the direct mapping method. The latter technique has been widely employed to solve problems with moving interfaces, and to study the asymptotics of the induced semiflows.
2013, 33(11&12): 5429-5440
doi: 10.3934/dcds.2013.33.5429
+[Abstract](1271)
+[PDF](145.3KB)
Abstract:
Of the many variations of vector measures, the Fréchet variation is finite valued but only subadditive. Finding a `controlling' finite measure for these in several cases, it is possible to develop a useful integration of the Bartle-Dunford-Schwartz type for many linear metric spaces. These include the generalized Orlicz spaces, $L^{\varphi}(\mu)$, where $\varphi$ is a concave $\varphi$-function with applications to stochastic measures $Z(\cdot)$ into various Fréchet spaces useful in prediction theory. In particular, certain $p$-stable random measures and a (sub) class of these leading to positive infinitely divisible ones are detailed.
Of the many variations of vector measures, the Fréchet variation is finite valued but only subadditive. Finding a `controlling' finite measure for these in several cases, it is possible to develop a useful integration of the Bartle-Dunford-Schwartz type for many linear metric spaces. These include the generalized Orlicz spaces, $L^{\varphi}(\mu)$, where $\varphi$ is a concave $\varphi$-function with applications to stochastic measures $Z(\cdot)$ into various Fréchet spaces useful in prediction theory. In particular, certain $p$-stable random measures and a (sub) class of these leading to positive infinitely divisible ones are detailed.
2013, 33(11&12): 5441-5455
doi: 10.3934/dcds.2013.33.5441
+[Abstract](1139)
+[PDF](408.0KB)
Abstract:
We propose a new method for estimating the eigenvalues of the thermal tensor of an anisotropically heat-conducting material, from transient thermal probe measurements of a heated thin cylinder.
We assume the principal axes of the thermal tensor to have been identified, and that the cylinder is oriented parallel to one of these axes (but we outline what is needed to overcome this limitation). The method involves estimating the first two Dirichlet eigenvalues (exponential decay rates) from transient thermal probe data. These implicitly determine the thermal diffusion coefficients (thermal tensor eigenvalues) in the directions of the other two axes. The process is repeated two more times with cylinders parallel to each of the remaining axes.
The method is tested by simulating a temperature probe time-series (obtained by solving the anisotropic heat equation numerically) and comparing the computed thermal tensor eigenvalues with their true values. The results are generally accurate to less than $1\%$ error.
We propose a new method for estimating the eigenvalues of the thermal tensor of an anisotropically heat-conducting material, from transient thermal probe measurements of a heated thin cylinder.
We assume the principal axes of the thermal tensor to have been identified, and that the cylinder is oriented parallel to one of these axes (but we outline what is needed to overcome this limitation). The method involves estimating the first two Dirichlet eigenvalues (exponential decay rates) from transient thermal probe data. These implicitly determine the thermal diffusion coefficients (thermal tensor eigenvalues) in the directions of the other two axes. The process is repeated two more times with cylinders parallel to each of the remaining axes.
The method is tested by simulating a temperature probe time-series (obtained by solving the anisotropic heat equation numerically) and comparing the computed thermal tensor eigenvalues with their true values. The results are generally accurate to less than $1\%$ error.
2013, 33(11&12): 5457-5491
doi: 10.3934/dcds.2013.33.5457
+[Abstract](1270)
+[PDF](557.4KB)
Abstract:
We obtain new insights into Hardy type Inequalities and the evolution problems associated to them. Surprisingly, the connection of the energy with the Hardy functionals is nontrivial, due to the presence of a Hardy singularity energy. This corresponds to a loss for the total energy. These problems are defined on bounded domains or the whole space.
We also consider equivalent problems with inverse square potential on exterior domains or the whole space. The extra energy term is then present as an effect that comes from infinity, a kind of hidden energy. In this case, in an unexpected way, this term is additive to the total energy, and it may even constitute the main part of it.
We obtain new insights into Hardy type Inequalities and the evolution problems associated to them. Surprisingly, the connection of the energy with the Hardy functionals is nontrivial, due to the presence of a Hardy singularity energy. This corresponds to a loss for the total energy. These problems are defined on bounded domains or the whole space.
We also consider equivalent problems with inverse square potential on exterior domains or the whole space. The extra energy term is then present as an effect that comes from infinity, a kind of hidden energy. In this case, in an unexpected way, this term is additive to the total energy, and it may even constitute the main part of it.
2013, 33(11&12): 5493-5506
doi: 10.3934/dcds.2013.33.5493
+[Abstract](1309)
+[PDF](411.6KB)
Abstract:
Let $A$ be a uniformly elliptic operator in divergence form with bounded coefficients. We show that on a bounded domain $Ω ⊂ \mathbb{R}^N$ with Lipschitz continuous boundary $∂Ω$, a realization of $Au-\beta_1(x,u)$ in $C(\bar{Ω})$ with the nonlinear general Wentzell boundary conditions $[Au-\beta_1(x,u)]|_{∂Ω}-\Delta_\Gamma u+\partial_\nu^au+\beta_2(x,u)= 0$ on $∂Ω$ generates a strongly continuous nonlinear semigroup on $C(\bar{Ω})$. Here, $\partial_\nu^au$ is the conormal derivative of $u$, and $\beta_1(x,\cdot)$ ($x \in Ω$), $\beta_2(x,\cdot)$ ($x \in ∂Ω$) are continuous on $\mathbb{R}$ satisfying a certain growth condition.
Let $A$ be a uniformly elliptic operator in divergence form with bounded coefficients. We show that on a bounded domain $Ω ⊂ \mathbb{R}^N$ with Lipschitz continuous boundary $∂Ω$, a realization of $Au-\beta_1(x,u)$ in $C(\bar{Ω})$ with the nonlinear general Wentzell boundary conditions $[Au-\beta_1(x,u)]|_{∂Ω}-\Delta_\Gamma u+\partial_\nu^au+\beta_2(x,u)= 0$ on $∂Ω$ generates a strongly continuous nonlinear semigroup on $C(\bar{Ω})$. Here, $\partial_\nu^au$ is the conormal derivative of $u$, and $\beta_1(x,\cdot)$ ($x \in Ω$), $\beta_2(x,\cdot)$ ($x \in ∂Ω$) are continuous on $\mathbb{R}$ satisfying a certain growth condition.
2013, 33(11&12): 5507-5519
doi: 10.3934/dcds.2013.33.5507
+[Abstract](1138)
+[PDF](348.9KB)
Abstract:
We discuss positive solutions of problems that arise from nonlinear boundary value problems in the particular situation where the nonlinear term $f(t,u)$ depends explicitly on $t$ and this dependence is crucial. We give new fixed point index results using comparisons with linear operators. These prove new results on existence of positive solutions under some conditions which can be sharp.
We discuss positive solutions of problems that arise from nonlinear boundary value problems in the particular situation where the nonlinear term $f(t,u)$ depends explicitly on $t$ and this dependence is crucial. We give new fixed point index results using comparisons with linear operators. These prove new results on existence of positive solutions under some conditions which can be sharp.
An example of large global smooth solution of 3 dimensional Navier-Stokes equations without pressure
2013, 33(11&12): 5521-5523
doi: 10.3934/dcds.2013.33.5521
+[Abstract](1288)
+[PDF](225.2KB)
Abstract:
We construct a global smooth solution of 3 dimensional Navier-Stokes equations in the torus, which also solves the heat equation. The solution is three dimensional and it can be arbitrarily large.
We construct a global smooth solution of 3 dimensional Navier-Stokes equations in the torus, which also solves the heat equation. The solution is three dimensional and it can be arbitrarily large.
2013, 33(11&12): 5525-5537
doi: 10.3934/dcds.2013.33.5525
+[Abstract](1542)
+[PDF](406.8KB)
Abstract:
In this paper, we consider the existence of weighted pseudo almost automorphic solutions of the semilinear integral equation $x(t)= \int_{-\infty}^{t}a(t-s)[Ax(s) + f(s,x(s))]ds, \ t\in\mathbb{R}$ in a Banach space $\mathbb{X}$, where $a\in L^{1}(\mathbb{R}_{+})$, $A$ is the generator of an integral resolvent family of linear bounded operators defined on the Banach space $\mathbb{X}$, and $f : \mathbb{R}\times\mathbb{X} \rightarrow \mathbb{X}$ is a weighted pseudo almost automorphic function. The main results are proved by using integral resolvent families, suitable composition theorems combined with the theory of fixed points.
In this paper, we consider the existence of weighted pseudo almost automorphic solutions of the semilinear integral equation $x(t)= \int_{-\infty}^{t}a(t-s)[Ax(s) + f(s,x(s))]ds, \ t\in\mathbb{R}$ in a Banach space $\mathbb{X}$, where $a\in L^{1}(\mathbb{R}_{+})$, $A$ is the generator of an integral resolvent family of linear bounded operators defined on the Banach space $\mathbb{X}$, and $f : \mathbb{R}\times\mathbb{X} \rightarrow \mathbb{X}$ is a weighted pseudo almost automorphic function. The main results are proved by using integral resolvent families, suitable composition theorems combined with the theory of fixed points.
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