Related papers: Existence of solutions for a fractional semilinear…

In this paper, we obtain necessary conditions and sufficient conditions on the initial data for the local-in-time solvability of the Cauchy problem \[ \partial_t u +(-\Delta)^\frac{\theta}{2} u=|x|^{-\gamma} u^p ,\quad x\in{\bf R}^N, t>0,…

Let $(u,v)$ be a solution to the Cauchy problem for a semilinear parabolic system \[ \mathrm{(P)} \qquad \cases{ \partial_t u=D_1\Delta u+v^p\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ \partial_t v=D_2\Delta v+u^q\quad &…

Let $(u,v)$ be a solution to a semilinear parabolic system \[ \mbox{(P)} \qquad \begin{cases} \partial_t u=D_1\Delta u+v^p\quad & \quad\mbox{in}\quad{\bf R}^N\times(0,T),\\ \partial_t v=D_2\Delta v+u^q\quad & \quad\mbox{in}\quad{\bf…

We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the…

Let $(u,v)$ be a nonnegative solution to the semilinear parabolic system \[ \mbox{(P)} \qquad \cases{ \partial_t u=D_1\Delta u+v^p, & $x\in{\bf R}^N,\,\,\,t>0,$\\ \partial_t v=D_2\Delta v+u^q, & $x\in{\bf R}^N,\,\,\,t>0,$\\…

In this paper we obtain necessary conditions and sufficient conditions on the initial data for the solvability of fractional semilinear heat equations with power nonlinearities in the Heisenberg group $\mathbb{H}^N$. Using these conditions,…

We establish the existence of solutions of the Cauchy problem for a higher-order semilinear parabolic equation by introducing a new majorizing kernel. We also study necessary conditions on the initial data for the existence of local-in-time…

We consider the Cauchy problem for the Hardy parabolic equation $\partial_t u-\Delta u=|x|^{-\gamma}u^p$ with initial data $u_0$ singular at some point $z$. Our main results show that, if $z\neq 0$, then the optimal strength of the…

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…

In this paper we consider the Cauchy problem for the semilinear damped wave equation $u_{tt}-\Delta u + u_t = h(u);\qquad u(0;x) = f(x); \quad u_t(0;x) = g(x);$ where $h(s) = |s|^{1+2/n}\mu(|s|)$. Here n is the space dimension and $\mu$ is…

We consider the Cauchy problem for a time fractional semilinear heat equation with initial data belonging to inhomogeneous/homogeneous Besov--Morrey spaces. We present sufficient conditions for the existence of local/global-in-time…

We study the Cauchy problem for the fractional Schr\"{o}dinger equation $$ i\partial_tu = (m^2-\Delta)^\frac\alpha2 u + F(u) in \mathbb{R}^{1+n}, $$ where $ n \ge 1$, $m \ge 0$, $1 < \alpha < 2$, and $F$ stands for the nonlinearity of…

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…

We study the existence and qualitative properties of solutions to the Cauchy problem associated to the quasilinear reaction-diffusion equation $$ \partial_tu=\Delta u^m+(1+|x|)^{\sigma}u^p, $$ posed for $(x,t)\in\real^N\times(0,\infty)$,…

We consider necessary conditions and sufficient conditions on the solvability of the Cauchy--Dirichlet problem for a fractional semilinear heat equation in open sets (possibly unbounded and disconnected) with a smooth boundary. Our…

We consider nonlinear parabolic equations involving fractional diffusion of the form $\partial_t u + (-\Delta)^s \Phi(u)= 0,$ with $0<s<1$, and solve an open problem concerning the existence of solutions for very singular nonlinearities…

We prove existence and uniqueness of Radon measure-valued 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 $u_0$…

Optimal conditions for initial data leading to non-existence of non-negative solutions to the Cauchy problem for the parabolic Hardy-H{\'e}non equation $$ \partial\_tu=\Delta u^m+|x|^{\sigma}u^p, \quad (t,x)\in(0,\infty)\times\mathbb{R}^N,…

We study the semilinear Cauchy problem for complex-valued damped evolution equations \begin{align*} \partial_t^2u+(-\Delta)^{\sigma}u+(-\Delta)^{\delta}\partial_tu=u^p,\ \ u(0,x)=u_0(x),\ \partial_tu(0,x)=u_1(x), \end{align*} with…

We prove the existence of a solution of (--$\Delta$) s u + f (u) = 0 in a smooth bounded domain $\Omega$ with a prescribed boundary value $\mu$ in the class of positive Radon measures for a large class of continuous functions f satisfying a…