相关论文: Prescribing Oscillation Behavior of Solutions to t…

In an open bounded real interval $\Omega$, the model for one-dimensional thermoelasticity given by \[ u_{tt} = u_{xx} - \big(f(\Theta)\big)_x, \qquad \Theta_t = \Theta_{xx} - f(\Theta) u_{xt}, \] is considered along with homogeneous…

Consider the following nonlinear one-dimensional stochastic fractional heat equation $$\frac{\partial }{\partial t}u(t, x)= -(-\Delta)^{\alpha/2}u(t, x) +\sigma(t,x,u(t,x)) \dot{W}(t, x), $$ where $-(-\Delta)^{\alpha/2}$ is the fractional…

We consider weak solutions of the fractional heat equation posed in the whole $n$-dimensional space, and establish their asymptotic convergence to the fundamental solution as $t\to\infty$ under the assumption that the initial datum is an…

We consider the semilinear heat equation $u_t - \Delta u = f(u)$ in $\Omega = B_R(0) \subset \mathbb{R}^n$ with super-exponential nonlinearities $f(u) = e^{u^p}u^q$ ($p>1$, $q \in \{0\}\cup [1,\infty)$), nonnegative bounded radially…

The standard problem for the classical heat equation posed in a bounded domain $\Omega$ of $\mathbb R^n$ is the initial and boundary value problem. If the Laplace operator is replaced by a version of the fractional Laplacian, the initial…

The goal of this paper is to obtain estimates for nonnegative solutions of the differential inequality $$\left(\frac{\partial}{\partial t} - \Delta\right) u \leq A u^p + B u $$ with small initial data in borderline Morrey norms over a…

In this work we consider a semi-linear energy critical wave equation in ${\mathbb R}^d$ ($3\leq d \leq 5$) \[ \partial_t^2 u - \Delta u = \pm \phi(x) |u|^{4/(d-2)} u, \qquad (x,t)\in {\mathbb R}^d \times {\mathbb R} \] with initial data…

In this paper we study existence of solutions for the Cauchy problem of the Debye-H\"{u}ckel system with low regularity initial data. By using the Chemin-Lerner time-space estimate for the heat equation, we prove that there exists a unique…

We establish the strong comparison principle and strict positivity of solutions to the following nonlinear stochastic heat equation on $\mathbb{R}^d$ \[ \left(\frac{\partial }{\partial t} -\frac{1}{2}\Delta \right) u(t,x) = \rho(u(t,x))…

We study a model that intermediates among the wave, heat, and transport equations. The approach considers the propagation of initial disturbances in a one-dimensional medium that can vibrate. The medium is nonlinear in such a form that…

This paper is concerned with the Cauchy problem for the semilinear wave equation: $u_{tt}-\Delta u=F(u) \ \mbox{in} \ R^n\times[0, \infty)$, where the space dimension $n \ge 2$, $F(u)=|u|^p$ or $F(u)=|u|^{p-1}u$ with $p>1$. Here, the Cauchy…

We analyse the behaviour of solutions of the linear heat equation in ${\mathbb R}^d$ for initial data in the classes $M_\varepsilon({\mathbb R}^d)$ of Radon measures with $\int_{{\mathbb R}^d}{\rm e}^{-\varepsilon|x|^2}\,{\rm…

This is a continuation, and conclusion, of our study of bounded solutions $u$ of the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line whose initial data $u_0=u(\cdot,0)$ have finite limits $\theta^\pm$ as $x\to\pm\infty$. We…

We consider the stochastic heat equation $\partial_{s}u =\frac{1}{2}\Delta u +(\beta V(s,y)-\lambda)u$, with a smooth space-time stationary Gaussian random field $V(s,y)$, in dimensions $d\geq 3$, with an initial condition…

We study the long time behavior of positive solutions of the Cauchy problem for nonlinear reaction-diffusion equations in $\mathbb{R}^N$ with bistable, ignition or monostable nonlinearities that exhibit threshold behavior. For $L^2$ initial…

In this paper we study the initial boundary value problem for the system $\mbox{div}(\sigma(u)\nabla\varphi)=0,\ \ u_t-\Delta u=\sigma(u)|\nabla\varphi|^2$. This problem is known as the thermistor problem which models the electrical heating…

When $P$ is the fractional Laplacian $(-\Delta )^a$, $0<a<1$, or a pseudodifferential generalization thereof, the Dirichlet problem for the associated heat equation over a smooth set $\Omega \subset{\Bbb R}^n$:…

We show the existence and the uniqueness of initial traces of nonnegative solutions to a semilinear heat equation on a half space of ${\mathbb R}^N$ under the zero Dirichlet boundary condition. Furthermore, we obtain necessary conditions…

This paper is devoted to studying the Cauchy problem for the Ostrovsky equation \begin{eqnarray*} \partial_{x}\left(u_{t}-\beta \partial_{x}^{3}u +\frac{1}{2}\partial_{x}(u^{2})\right) -\gamma u=0, \end{eqnarray*} with positive $\beta$ and…

We study nonnegative solutions of the Cauchy problem $$ \begin{cases} u_t+[\varphi(u)]_x=0 & \text{in } \mathbb{R}\times (0,T) \\ u=u_0\ge 0&\text{in } \mathbb{R}\times \{0\}, \end{cases} $$ where $u_0$ is a Radon measure and…