American Institute of Mathematical Sciences

In this paper, we investigate the pointwise time analyticity of three differential equations. They are the biharmonic heat equation, the heat equation with potentials and some nonlinear heat equations with power nonlinearity of order \begin{document}$ p $\end{document}. The potentials include all the nonnegative ones. For the first two equations, we prove if \begin{document}$ u $\end{document} satisfies some growth conditions in \begin{document}$ (x,t)\in \mathrm{M}\times [0,1] $\end{document}, then \begin{document}$ u $\end{document} is analytic in time \begin{document}$ (0,1] $\end{document}. Here \begin{document}$ \mathrm{M} $\end{document} is \begin{document}$ R^d $\end{document} or a complete noncompact manifold with Ricci curvature bounded from below by a constant. Then we obtain a necessary and sufficient condition such that \begin{document}$ u(x,t) $\end{document} is analytic in time at \begin{document}$ t = 0 $\end{document}. Applying this method, we also obtain a necessary and sufficient condition for the solvability of the backward equations, which is ill-posed in general.

For the nonlinear heat equation with power nonlinearity of order \begin{document}$ p $\end{document}, we prove that a solution is analytic in time \begin{document}$ t\in (0,1] $\end{document} if it is bounded in \begin{document}$ \mathrm{M}\times[0,1] $\end{document} and \begin{document}$ p $\end{document} is a positive integer. In addition, we investigate the case when \begin{document}$ p $\end{document} is a rational number with a stronger assumption \begin{document}$ 0<C_3 \leq |u(x,t)| \leq C_4 $\end{document}. It is also shown that a solution may not be analytic in time if it is allowed to be \begin{document}$ 0 $\end{document}. As a lemma, we obtain an estimate of \begin{document}$ \partial_t^k \Gamma(x,t;y) $\end{document} where \begin{document}$ \Gamma(x,t;y) $\end{document} is the heat kernel on a manifold, with an explicit estimation of the coefficients.

An interesting point is that a solution may be analytic in time even if it is not smooth in the space variable \begin{document}$ x $\end{document}, implying that the analyticity of space and time can be independent. Besides, for general manifolds, space analyticity may not hold since it requires certain bounds on curvature and its derivatives.

In this paper, a generalitzation of the \begin{document}$ L_{p} $\end{document}-Christoffel-Minkowski problem is studied. We consider an anisotropic curvature flow and derive the long-time existence of the flow. Then under some initial data, we obtain the existence of smooth solutions to this problem for \begin{document}$ c = 1 $\end{document}.

In this paper, we consider a family of parabolic systems with singular nonlinearities. We study the classification of global existence and quenching of solutions according to parameters and initial data. Furthermore, the rate of the convergence of the global solutions to the minimal steady state is given. Due to the lack of variational characterization of the first eigenvalue to the linearized elliptic problem associated with our parabolic system, some new ideas and techniques are introduced.

We study a class of nonlocal partial differential equations with nonlinear perturbations, which is a general model for some equations arose from fluid dynamics. Our aim is to analyze some sufficient conditions ensuring the global solvability, regularity and stability of solutions. Our analysis is based on the theory of completely positive kernel functions, local estimates and a new Gronwall type inequality.

In this paper, we investigate the nonnegative solutions of the nonlinear singular integral system

\begin{document}$ \begin{equation} \left\{ \begin{array}{lll} u_i(x) = \int_{\mathbb{R}^n}\frac{1}{|x-y|^{n-\alpha}|y|^{a_i}}f_i(u(y))dy,\quad x\in\mathbb{R}^n,\quad i = 1,2\cdots,m,\\ 0<\alpha<n,\quad u(x) = (u_1(x),\cdots,u_m(x)),\nonumber \end{array}\right. \end{equation} $\end{document}

where \begin{document}$ 0<a_i/2<\alpha $\end{document}, \begin{document}$ f_i(u) $\end{document}, \begin{document}$ 1\leq i\leq m $\end{document}, are real-valued functions, nonnegative and monotone nondecreasing with respect to the independent variables \begin{document}$ u_1 $\end{document}, \begin{document}$ u_2 $\end{document}, \begin{document}$ \cdots $\end{document}, \begin{document}$ u_m $\end{document}. By the method of moving planes in integral forms, we show that the nonnegative solution \begin{document}$ u = (u_1,u_2,\cdots,u_m) $\end{document} is radially symmetric when \begin{document}$ f_i $\end{document} satisfies some monotonicity condition.

We present asymptotic analysis of Couette flow through a channel packed with porous medium. We assume that the porous medium is anisotropic and the permeability varies along all the directions so that it appears as a positive semidefinite matrix in the momentum equation. We developed existence and uniqueness results corresponding to the anisotropic Brinkman-Forchheimer extended Darcy's equation in case of fully developed flow using the Browder-Minty theorem. Complemented with the existence and uniqueness analysis, we present an asymptotic solution by taking Darcy number as the perturbed parameter. For a high Darcy number, the corresponding problem is dealt with regular perturbation expansion. For low Darcy number, the problem of interest is a singular perturbation. We use matched asymptotic expansion to treat this case. More generally, we obtained an approximate solution for the nonlinear problem, which is uniformly valid irrespective of the porous medium parameter values. The analysis presented serves a dual purpose by providing the existence and uniqueness of the anisotropic nonlinear Brinkman-Forchheimer extended Darcy's equation and provide an approximate solution that shows good agreement with the numerical solution.

We investigate Hardy-Rellich inequalities for perturbed Laplacians. In particular, we show that a non-trivial angular perturbation of the free operator typically improves the inequality, and may also provide an estimate which does not hold in the free case. The main examples are related to the introduction of a magnetic field: this is a manifestation of the diamagnetic phenomenon, which has been observed by Laptev and Weidl in [21] for the Hardy inequality, later by Evans and Lewis in [9] for the Rellich inequality; however, to the best of our knowledge, the so called Hardy-Rellich inequality has not yet been investigated in this regards. After showing the optimal inequality, we prove that the best constant is not attained by any function in the domain of the estimate.

We classify the finite time blow-up profiles for the following reaction-diffusion equation with unbounded weight:

\begin{document}$ \partial_tu = \Delta u^m+|x|^{\sigma}u^p, $\end{document}

posed in any space dimension \begin{document}$ x\in \mathbb{R}^N $\end{document}, \begin{document}$ t\geq0 $\end{document} and with exponents \begin{document}$ m>1 $\end{document}, \begin{document}$ p\in(0, 1) $\end{document} and \begin{document}$ \sigma>2(1-p)/(m-1) $\end{document}. We prove that blow-up profiles in backward self-similar form exist for the indicated range of parameters, showing thus that the unbounded weight has a strong influence on the dynamics of the equation, merging with the nonlinear reaction in order to produce finite time blow-up. We also prove that all the blow-up profiles are compactly supported and might present two different types of interface behavior and three different possible good behaviors near the origin, with direct influence on the blow-up behavior of the solutions. We classify all these profiles with respect to these different local behaviors depending on the magnitude of \begin{document}$ \sigma $\end{document}. This paper generalizes in dimension \begin{document}$ N>1 $\end{document} previous results by the authors in dimension \begin{document}$ N = 1 $\end{document} and also includes some finer classification of the profiles for \begin{document}$ \sigma $\end{document} large that is new even in dimension \begin{document}$ N = 1 $\end{document}.

In this paper, we consider the following haptotaxis model describing cancer cells invasion and metastatic spread

\begin{document}$\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{l}}{{u_t} = \Delta u - \chi \nabla \cdot (u\nabla w),}&{x \in \Omega ,\;t > 0,}\\{{v_t} = {d_v}\Delta v - \xi \nabla \cdot (v\nabla w),}&{x \in \Omega ,\;t > 0,}\\{{m_t} = {d_m}\Delta m + u - m,}&{x \in \Omega ,\;t > 0,}\\{{w_t} = - \left( {{\gamma _1}u + m} \right)w,}&{x \in \Omega ,\;t > 0,}\end{array}} \right.}&{(0.1)}\end{array}$ \end{document}

where \begin{document}$ \Omega\subset \mathbb{R}^3 $\end{document} is a bounded domain with smooth boundary and the parameters \begin{document}$ \chi, \xi, d_{v}, d_{m},\gamma_{1}>0 $\end{document}. Under homogeneous boundary conditions of Neumann type for \begin{document}$ u $\end{document}, \begin{document}$ v $\end{document}, \begin{document}$ m $\end{document} and \begin{document}$ w $\end{document}, it is proved that, for suitable smooth initial data \begin{document}$ (u_0, v_0, m_0, w_0) $\end{document}, the corresponding Neumann initial-boundary value problem possesses a global generalized solution.

We study the existence of an inertial manifold for the solutions to fully non-autonomous parabolic differential equation of the form

\begin{document}$ \dfrac{du}{dt} + A(t)u(t) = f(t,u),\, t> s. $\end{document}

We prove the existence of such an inertial manifold in the case that the family of linear partial differential operators \begin{document}$ (A(t))_{t\in { \mathbb {R}}} $\end{document} generates an evolution family \begin{document}$ (U(t,s))_{t\ge s} $\end{document} satisfying certain dichotomy estimates, and the nonlinear forcing term \begin{document}$ f(t,x) $\end{document} satisfies the Lipschitz condition such that certain dichotomy gap condition holds.

Let \begin{document}$ n\ge2 $\end{document} and \begin{document}$ \Omega\subset\mathbb{R}^n $\end{document} be a bounded NTA domain. In this article, the authors study (weighted) global regularity estimates for Neumann boundary value problems of second-order elliptic equations of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in \begin{document}$ \Omega $\end{document}. Precisely, for any given \begin{document}$ p\in(2,\infty) $\end{document}, via a weak reverse Hölder inequality with the exponent \begin{document}$ p $\end{document}, the authors give a sufficient condition for the global \begin{document}$ W^{1,p} $\end{document} estimate and the global weighted \begin{document}$ W^{1,q} $\end{document} estimate, with \begin{document}$ q\in[2,p] $\end{document} and some Muckenhoupt weights, of solutions to Neumann boundary value problems in \begin{document}$ \Omega $\end{document}. As applications, the authors further obtain global regularity estimates for solutions to Neumann boundary value problems of second-order elliptic equations of divergence form with coefficients consisting of both a small \begin{document}$ \mathrm{BMO} $\end{document} symmetric part and a small \begin{document}$ \mathrm{BMO} $\end{document} anti-symmetric part, respectively, in bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, \begin{document}$ C^1 $\end{document} domains, or (semi-)convex domains, in weighted Lebesgue spaces. The results given in this article improve the known results by weakening the assumption on the coefficient matrix.

In the present paper we study the asymptotic behavior of discretized finite dimensional dynamical systems. We prove that under some discrete angle condition and under a Lojasiewicz's inequality condition, the solutions to an implicit scheme converge to equilibrium points. We also present some numerical simulations suggesting that our results may be extended under weaker assumptions or to infinite dimensional dynamical systems.

By means of the method of moving planes, we study the monotonicity of positive solutions to degenerate quasilinear elliptic systems in half-spaces. We also prove the symmetry of positive solutions to the systems in strips by using similar arguments. Our work extends the main results obtained in [16,20] to the system, in which substantial differences with the single cases are presented.

This paper is devoted to the construction of analogues of higher Ekeland-Hofer symplectic capacities for \begin{document}$ P $\end{document}-symmetric subsets in the standard symplectic space \begin{document}$ (\mathbb{R}^{2n},\omega_0) $\end{document}, which is motivated by Long and Dong's study about \begin{document}$ P $\end{document}-symmetric closed characteristics on \begin{document}$ P $\end{document}-symmetric convex bodies. We study the relationship between these capacities and other capacities, and give some computation examples. Moreover, we also define higher real symmetric Ekeland-Hofer capacities as a complement of Jin and the second named author's recent study of the real symmetric analogue about the first Ekeland-Hofer capacity.

In this paper, we discuss the generalized quasilinear Schrödinger equation with Kirchhoff-type:

\begin{document}$\left (1\!+\!b\int_{\mathbb{R}^{3}}g^{2}(u)|\nabla u|^{2} dx \right) \left[-\mathrm{div} \left(g^{2}(u)\nabla u\right)\!+\!g(u)g'(u)|\nabla u|^{2}\right] \!+\!V(x)u\! = \!f( u),(\rm P)$ \end{document}

where \begin{document}$ b>0 $\end{document} is a parameter, \begin{document}$ g\in \mathbb{C}^{1}(\mathbb{R},\mathbb{R}^{+}) $\end{document}, \begin{document}$ V\in \mathbb{C}^{1}(\mathbb{R}^3,\mathbb{R}) $\end{document} and \begin{document}$ f\in \mathbb{C}(\mathbb{R},\mathbb{R}) $\end{document}. Under some "Berestycki-Lions type assumptions" on the nonlinearity \begin{document}$ f $\end{document} which are almost necessary, we prove that problem \begin{document}$ (\rm P) $\end{document} has a nontrivial solution \begin{document}$ \bar{u}\in H^{1}(\mathbb{R}^{3}) $\end{document} such that \begin{document}$ \bar{v} = G(\bar{u}) $\end{document} is a ground state solution of the following problem

\begin{document}$-\left(1+b\int_{\mathbb{R}^{3}} |\nabla v|^{2} dx \right) \triangle v+V(x)\frac{G^{-1}(v)}{g(G^{-1}(v))} = \frac{f(G^{-1}(v))}{g(G^{-1}(v))},(\rm \bar{P})$ \end{document}

where \begin{document}$ G(t): = \int_{0}^{t} g(s) ds $\end{document}. We also give a minimax characterization for the ground state solution \begin{document}$ \bar{v} $\end{document}.

In this paper, we introduce frames with the uniform approximation property (UAP). We show that with a UAP frame, it is efficient to recover a signal from its frame coefficients with one erasure while the recovery error is smaller than that with the ordinary recovery algorithm. In fact, our approach gives a balance between the recovery accuracy and the computational complexity.