相关论文: On the Fujita exponent for a Hardy-H\'{e}non equat…

In this paper, we study the semilinear heat equation with a forcing term, driven by the fractional sub-Laplacian (-\Delta_{\mathbbm{H}^N})^s of order $s\in (0,1),$ on the Heisenberg group $\mathbbm{H}^N$. We establish that the Fujita…

The aim of this paper is to treat the following problem $$ (P) \left\{ \begin{array}{rcll} (-\Delta)^s_{p, \beta} u &= & f(x,u) &\mbox{ in }\Omega, u & = & 0 &\mbox{ in } \mathds{R}^N\setminus\Omega, \end{array} \right. $$ where $$…

In this paper we consider the problem: $\partial_{t} u- \Delta u=f(u),\; u(0)=u_0\in \exp L^p(\R^N),$ where $p>1$ and $f : \R\to\R$ having an exponential growth at infinity with $f(0)=0.$ We prove local well-posedness in $\exp L^p_0(\R^N)$…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a bounded smooth domain and $\delta(x)=\text{dist}(x,\partial \Omega)$. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to $$…

We investigate nonexistence of nontrivial nonnegative solutions to a class of semilinear parabolic equations with a positive potential, posed on weighted graphs. Assuming an upper bound on the Laplacian of the distance and a suitable…

The global existence for semilinear wave equations with space-dependent critical damping $\partial_t^2u-\Delta u+\frac{V_0}{|x|}\partial_t u=f(u)$ in an exterior domain is dealt with, where $f(u)=|u|^{p-1}u$ and $f(u)=|u|^p$ are in mind.…

We study the well-posedness of a non-linear heat equation with power nonlinearity with positive initial data on quantum Euclidean spaces. We prove a noncommutative analogue of the classical Fujita theorem by identifying the critical…

If $p>1+2/n$ then the equation $u_t-\Delta u = u^p, \quad x\in{\mathbb R}^n,\ t>0,$ possesses both positive global solutions and positive solutions which blow up in finite time. We study the large time behavior of radial positive solutions…

Global self-similar solutions to the parabolic Hardy-H\'enon equation $$ u_t=\Delta u^m+|x|^{\sigma}u^p, \quad (x,t)\in\mathbb{R}^N\times(0,\infty), $$ are classified in the range of exponents $m\geq1$, $p>m$ and $\sigma>\max\{-2,-N\}$. The…

We consider nonlinear parabolic SPDEs of the form $\partial_t u=\sL u + \sigma(u)\dot w$, where $\dot w$ denotes space-time white noise, $\sigma:\R\to\R$ is [globally] Lipschitz continuous, and $\sL$ is the $L^2$-generator of a L\'evy…

The aim of this paper is to give global nonexistence and blow--up results for the problem $$ \begin{cases} u_{tt}-\Delta u+P(x,u_t)=f(x,u) \qquad &\text{in $(0,\infty)\times\Omega$,}\\ u=0 &\text{on $(0,\infty)\times \Gamma_0$,}\\…

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 are concerned with the following time-fractional semilinear heat equation in the $N$-dimensional whole space ${\bf R}^N$ with $N \geq 1$. \[ {\rm (P)}_\alpha \qquad \partial_t^\alpha u -\Delta u = u^p,\quad t>0,\,\,\, x\in{\bf R}^N,…

In this work, we are interested on the study of the Fujita exponent and the meaning of the blow-up for the Fractional Cauchy problem with the Hardy potential, namely, \begin{equation*} u_t+(-\Delta)^s…

This work investigates the semilinear wave equation featuring the displacement dependent term $\sigma(u)\partial_t u $ and nonlinearity $f(u)$. By developing refined space-time a priori estimates under extended ranges of the nonlinearity…

In this paper, we study the local behavior of nonnegative solutions of fractional semi-linear equations $(-\Delta)^\sigma u = u^p$ with an isolated singularity, where $\sg \in (0, 1)$ and $\frac{n}{n-2\sg} < p < \frac{n+2\sg}{n-2\sg}$. We…

In this paper, we derive sufficient conditions on initial data for the local-in-time solvability of a time-fractional semilinear heat equation with the Fujita exponent in a uniformly local weak Zygmund type space. It is known that the…

In this article, we study the space-time SPDE $$ \partial_t^\beta u=-(-\Delta)^{\alpha/2} u+I_t^{1-\beta}[b(u)+\sigma(u)\dot{W}],$$ where $u=u(t,x)$ is defined for $(t,x)\in\mathbb{R}_+\times \mathbb{R},$ $\beta\in(0,1), \alpha\in(0,2)$ and…

Consider the equation u_t=\Delta u-Vu +au^p \text{in} R^n\times (0,T); u(x,0)=\phi(x)\gneq0, \text{in} R^n, where $p>1$, $n\ge2$, $T\in(0,\infty]$, $V(x)\sim\frac\omega{|x|^2}$ as $|x|\to\infty$, for some $\omega\neq0$, and $a(x)$ is on the…

In this paper we examine well-posedness for a class of fourth-order nonlinear parabolic equation $\partial_t u + (-\Delta)^2 u = \nabla \cdot F(\nabla u)$, where $F$ satisfies a cubic growth conditions. We establish existence and uniqueness…