# American Institute of Mathematical Sciences

eISSN:
2688-1594

All Issues

## Electronic Research Archive

### Special issue on PDEs arising from nonlinear waves and fluid dynamics

Robin Ming Chen, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA mingchen@pitt.edu

Runzhang Xu, College of Mathematical Sciences, Harbin Engineering University, China xurunzh@163.com

2020, 28(1): 15-25 doi: 10.3934/era.2020002 +[Abstract](178)+[HTML](94) +[PDF](327.55KB)
Abstract:

In this paper, we consider the dissipative Degasperis-Procesi equation with arbitrary dispersion coefficient and compactly supported initial data. We establish the simple condition on the initial data which lead to guarantee that the solution exists globally. We also investigate the propagation speed for the equation under the initial data is compactly supported.

2020, 28(1): 27-46 doi: 10.3934/era.2020003 +[Abstract](168)+[HTML](72) +[PDF](346.98KB)
Abstract:

In this paper, the Cauchy problem of the \begin{document}$3$\end{document}D compressible Navier-Stokes equations with degenerate viscosities and far field vacuum is considered. We prove that the \begin{document}$L^\infty$\end{document} norm of the deformation tensor \begin{document}$D(u)$\end{document} (\begin{document}$u$\end{document}: the velocity of fluids) and the \begin{document}$L^6$\end{document} norm of \begin{document}$\nabla \log \rho$\end{document} (\begin{document}$\rho$\end{document}: the mass density) control the possible blow-up of regular solutions. This conclusion means that if a solution with far field vacuum to the Cauchy problem of the compressible Navier-Stokes equations with degenerate viscosities is initially regular and loses its regularity at some later time, then the formation of singularity must be caused by losing the bound of \begin{document}$D(u)$\end{document} or \begin{document}$\nabla \log \rho$\end{document} as the critical time approaches; equivalently, if both \begin{document}$D(u)$\end{document} and \begin{document}$\nabla \log \rho$\end{document} remain bounded, a regular solution persists.

2020, 28(1): 47-66 doi: 10.3934/era.2020004 +[Abstract](145)+[HTML](75) +[PDF](413.61KB)
Abstract:

This paper is concerned with the initial boundary value problem for a shear thinning fluid-particle interaction non-Newtonian model with vacuum. The viscosity term of the fluid and the non-Newtonian gravitational force are fully nonlinear. Under Dirichlet boundary for velocity and the no-flux condition for density of particles, the existence and uniqueness of strong solutions is investigated in one dimensional bounded intervals.

2020, 28(1): 67-90 doi: 10.3934/era.2020005 +[Abstract](177)+[HTML](93) +[PDF](777.08KB)
Abstract:

This paper deals with the global existence and blow-up of solutions to a inhomogeneous pseudo-parabolic equation with initial value \begin{document}$u_0$\end{document} in the Sobolev space \begin{document}$H_0^1( \Omega)$\end{document}, where \begin{document}$\Omega\subset \mathbb{R}^n$\end{document} (\begin{document}$n\geq1$\end{document} is an integer) is a bounded domain. By using the mountain-pass level \begin{document}$d$\end{document} (see (14)), the energy functional \begin{document}$J$\end{document} (see (12)) and Nehari function \begin{document}$I$\end{document} (see (13)), we decompose the space \begin{document}$H_0^1( \Omega)$\end{document} into five parts, and in each part, we show the solutions exist globally or blow up in finite time. Furthermore, we study the decay rates for the global solutions and lifespan (i.e., the upper bound of blow-up time) of the blow-up solutions. Moreover, we give a blow-up result which does not depend on \begin{document}$d$\end{document}. By using this theorem, we prove the solution can blow up at arbitrary energy level, i.e. for any \begin{document}$M\in \mathbb{R}$\end{document}, there exists \begin{document}$u_0\in H_0^1( \Omega)$\end{document} satisfying \begin{document}$J(u_0) = M$\end{document} such that the corresponding solution blows up in finite time.

2020, 28(1): 91-102 doi: 10.3934/era.2020006 +[Abstract](181)+[HTML](79) +[PDF](299.33KB)
Abstract:

We study the initial boundary value problem of linear homogeneous wave equation with dynamic boundary condition. We aim to prove the finite time blow-up of the solution at critical energy level or high energy level with the nonlinear damping term on boundary in control systems.

2020, 28(1): 221-261 doi: 10.3934/era.2020015 +[Abstract](17)+[HTML](10) +[PDF](498.76KB)
Abstract:

In this paper, we consider the initial boundary value problem for the fourth order wave equation with nonlinear boundary velocity feedbacks \begin{document}$f_{1}(u_{\nu{t}})$\end{document}, \begin{document}$f_{2}(u_{t})$\end{document} and internal source \begin{document}$|u|^{\rho}u$\end{document}. Under some geometrical conditions, the existence and uniform decay rates of the solutions are proved even if the nonlinear boundary velocity feedbacks \begin{document}$f_{1}(u_{\nu{t}})$\end{document}, \begin{document}$f_{2}(u_{t})$\end{document} have not polynomial growth near the origin respectively. By the combination of the Galerkin approximation, potential well method and a special basis constructed, we first obtain the global existence and uniqueness of regular solutions and weak solutions. In addition, we also investigate the explicit decay rate estimates of the energy, the ideas of which are based on the construction of a special weight function \begin{document}$\phi(t)$\end{document} (that depends on the behaviors of the functions \begin{document}$f_{1}(u_{\nu{t}})$\end{document}, \begin{document}$f_{2}(u_{t})$\end{document} near the origin), nonlinear integral inequality and the Multiplier method.

2020, 28(1): 263-289 doi: 10.3934/era.2020016 +[Abstract](11)+[HTML](8) +[PDF](383.54KB)
Abstract:

In this paper, we consider a plate equation with nonlinear damping and logarithmic source term. By the contraction mapping principle, we establish the local existence. The global existence and decay estimate of the solution at subcritical initial energy are obtained. We also prove that the solution with negative initial energy blows up in finite time under suitable conditions. Moreover, we also give the blow-up in finite time of solution at the arbitrarily high initial energy for linear damping (i.e. \begin{document}$m = 2$\end{document}).

2020, 28(1): 291-309 doi: 10.3934/era.2020017 +[Abstract](13)+[HTML](5) +[PDF](418.12KB)
Abstract:

In this paper, we prove the existence of positive solutions with prescribed \begin{document}$L^{2}$\end{document}-norm to the following Choquard equation:

where \begin{document}$\lambda\in \mathbb{R}, \alpha\in (0,3)$\end{document} and \begin{document}$I_{\alpha}: \mathbb{R}^3\rightarrow \mathbb{R}$\end{document} is the Riesz potential. Under the weaker conditions, by using a minimax procedure and some new analytical techniques, we show that for any \begin{document}$c>0$\end{document}, the above equation possesses at least a couple of weak solution \begin{document}$(\bar{u}_c, \bar{ \lambda}_c)\in \mathcal{S}_{c}\times \mathbb{R}^-$\end{document} such that \begin{document}$\|\bar{u}_c\|_{2}^{2} = c$\end{document}.

2020, 28(1): 311-326 doi: 10.3934/era.2020018 +[Abstract](12)+[HTML](5) +[PDF](371.44KB)
Abstract:

This paper is concerned with the initial-boundary value problem for a class of viscoelastic plate equations on an arbitrary dimensional bounded domain. Under certain assumptions on the memory kernel and the source term, the global well-posedness of solutions and the existence of global attractors are obtained.

2020, 28(1): 327-346 doi: 10.3934/era.2020019 +[Abstract](16)+[HTML](9) +[PDF](358.13KB)
Abstract:

In this paper, some criteria for weakly approximative compactness and approximative compactness of weak\begin{document}$^{*}$\end{document} hyperplane for Musielak-Orlicz-Bochner function spaces are given. Moreover, we also prove that, in Musielak-Orlicz-Bochner function spaces generated by strongly smooth Banach space, \begin{document}$L_{M}^{0}(X)$\end{document} (resp \begin{document}$L_{M}(X)$\end{document}) is an Asplund space if and only if \begin{document}$M$\end{document} and \begin{document}$N$\end{document} satisfy condition \begin{document}$\Delta$\end{document}. As a corollary, we obtain that \begin{document}$L_{M}^{0}(R)$\end{document} (resp \begin{document}$L_{M}(R)$\end{document}) is an Asplund space if and only if \begin{document}$M$\end{document} and \begin{document}$N$\end{document} satisfy condition \begin{document}$\Delta$\end{document}.

2020, 28(1): 347-367 doi: 10.3934/era.2020020 +[Abstract](17)+[HTML](14) +[PDF](404.01KB)
Abstract:

As a typical example of our analysis we consider a generalized Boussinesq equation, linearly damped and with a nonlinear source term. For any positive value of the initial energy, in particular for high energies, we give sufficient conditions on the initial data to conclude nonexistence of global solutions. We do our analysis in an abstract framework. We compare our results with those in the literature and we give more examples to illustrate the applicability of the abstract formulation.

2020, 28(1): 369-381 doi: 10.3934/era.2020021 +[Abstract](16)+[HTML](6) +[PDF](319.98KB)
Abstract:

For a class of semilinear parabolic equations under nonlinear dynamical boundary conditions in a bounded domain, we obtain finite time blow-up solutions when the initial data varies in the phase space \begin{document}$H_0^1(\Omega)$\end{document} at positive initial energy level and get global solutions with the initial data at low and critical energy level. Our main tools are potential well method and concavity method.

2020, 28(1): 383-404 doi: 10.3934/era.2020022 +[Abstract](11)+[HTML](8) +[PDF](395.33KB)
Abstract:

This paper is concerned with the following Schrödinger-Poisson system

where \begin{document}$p\in (3,5)$\end{document}, \begin{document}$K(x)$\end{document} and \begin{document}$h(x)$\end{document} are nonnegative functions, and \begin{document}$\mu$\end{document} is a positive parameter. Let \begin{document}$\mu_1 > 0$\end{document} be an isolated first eigenvalue of the eigenvalue problem \begin{document}$-\Delta u + u = \mu h(x)u$\end{document}, \begin{document}$u\in H^1(\mathbb{R}^3)$\end{document}. As \begin{document}$0<\mu\leq\mu_1$\end{document}, we prove that \begin{document}$(P_{\mu})$\end{document} has at least one nonnegative bound state with positive energy. As \begin{document}$\mu > \mu_1$\end{document}, there is \begin{document}$\delta > 0$\end{document} such that for any \begin{document}$\mu\in (\mu_1, \mu_1 + \delta)$\end{document}, \begin{document}$(P_\mu)$\end{document} has a nonnegative ground state \begin{document}$u_{0,\mu}$\end{document} with negative energy, and \begin{document}$u_{0,\mu^{(n)}}\to 0$\end{document} in \begin{document}$H^1(\mathbb{R}^3)$\end{document} as \begin{document}$\mu^{(n)}\downarrow \mu_1$\end{document}. Besides, \begin{document}$(P_\mu)$\end{document} has another nonnegative bound state \begin{document}$u_{2,\mu}$\end{document} with positive energy, and \begin{document}$u_{2,\mu^{(n)}}\to u_{\mu_1}$\end{document} in \begin{document}$H^1(\mathbb{R}^3)$\end{document} as \begin{document}$\mu^{(n)}\downarrow \mu_1$\end{document}, where \begin{document}$u_{\mu_1}$\end{document} is a bound state of \begin{document}$(P_{\mu_1})$\end{document}.

2020, 28(1): 433-457 doi: 10.3934/era.2020025 +[Abstract](21)+[HTML](9) +[PDF](421.49KB)
Abstract:

In this paper, we consider the fourth-order Moore-Gibson- Thompson equation with memory recently introduced by (Milan J. Math. 2017, 85: 215-234) that proposed the fourth-order model. We discuss the well-posedness of the solution by using Faedo-Galerkin method. On the other hand, for a class of relaxation functions satisfying \begin{document}$g'(t)\leq-\xi(t)M(g(t))$\end{document} for \begin{document}$M$\end{document} to be increasing and convex function near the origin and \begin{document}$\xi(t)$\end{document} to be a nonincreasing function, we establish the explicit and general energy decay result, from which we can improve the earlier related results.

2020, 28(2): 627-649 doi: 10.3934/era.2020033 +[Abstract](6)+[HTML](2) +[PDF](425.86KB)
Abstract:

In this paper, our goal is to improve the local well-posedness theory for certain generalized Boussinesq equations by revisiting bilinear estimates related to the Schrödinger equation. Moreover, we propose a novel, automated procedure to handle the summation argument for these bounds.

2020, 28(2): 651-669 doi: 10.3934/era.2020034 +[Abstract](6)+[HTML](2) +[PDF](429.17KB)
Abstract:

The aim of this paper is to investigate the existence of weak solutions for a Kirchhoff-type differential inclusion wave problem involving a discontinuous set-valued term, the fractional \begin{document}$p$\end{document}-Laplacian and linear strong damping term. The existence of weak solutions is obtained by using a regularization method combined with the Galerkin method.

2020, 28(2): 671-689 doi: 10.3934/era.2020035 +[Abstract](7)+[HTML](5) +[PDF](361.94KB)
Abstract:

Nonlinear Klein-Gordon equation with combined power type nonlinearity and critical initial energy is investigated. The qualitative properties of a new ordinary differential equation are studied and the concavity method of Levine is improved. Necessary and sufficient conditions for finite time blow up and global existence of the solutions are proved. New sufficient conditions on the initial data for finite time blow up, based on the necessary and sufficient ones, are obtained. The asymptotic behavior of the global solutions is also investigated.

2020, 28(2): 691-719 doi: 10.3934/era.2020036 +[Abstract](4)+[HTML](4) +[PDF](457.43KB)
Abstract:

In this paper, a rotating blades equation is considered. The arbitrary pre-twisted angle, arbitrary pre-setting angle and arbitrary rotating speed are taken into account when establishing the rotating blades model. The nonlinear PDEs of motion and two types of boundary conditions are derived by the extended Hamilton principle and the first-order piston theory. The well-posedness of weak solution (global in time) for the rotating blades equation with Clamped-Clamped (C-C) boundary conditions can be proved by compactness method and energy method. Strong energy estimates are derived under additional assumptions on the initial data. In addition, the existence and regularity of weak solutions (global in time) for the rotating blades equation with Clamped-Free (C-F) boundary conditions are proved as well.

2020,  doi: 10.3934/era.2020109 +[Abstract](15)+[HTML](11) +[PDF](264.92KB)
Abstract:

In this paper, we consider a kind of efficient finite difference methods (FDMs) for solving the nonlinear Helmholtz equation in the Kerr medium. Firstly, by applying several iteration methods, we linearize the nonlinear Helmholtz equation in several different ways. Then, based on the resulted linearized problem at each iterative step, by rearranging the Taylor expansion and using the ADI method, we deduce a kind of new FDMs, which also provide a route to deal with the problem with discontinuous coefficients.Finally, some numerical results are presented to validate the efficiency of the proposed schemes, and to show that our schemes perform with much higher accuracy and better convergence compared with the classical ones.

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