Related papers: Nonexistence of solutions to parabolic problems wi…
This paper studies the properties of solutions to a class of elliptic and parabolic problems involving the fractional Laplacian. By applying the mountain pass theorem, we prove the existence of bounded classical positive solutions in the…
We are concerned with the Cauchy problem for the semilinear parabolic equation driven by the mixed local-nonlocal operator $\mathcal{L} = -\Delta+(-\Delta)^s$, with a power-like source term. We show that the so-called Fujita phenomenon…
In this paper we study one dimensional parabolic free boundary value problem with a nonlocal (integro-differential) condition on the free boundary. We establish global existence-uniqueness of classical solutions assuming that the…
We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solutions which grow at most like the distance to the boundary to a power given by the natural scaling exponent of the equation; in other words,…
We consider the semilinear heat equation $$ u_t-\Delta u=|u|^{p-1}u,\ \ (t,x)\in\mathbb{R}^+\times\mathbb{R}^n. $$ The well-known difficulty with this problem is that the potential well method cannot be applied directly, due to the scaling…
We consider the Cauchy problem for a system of fully nonlinear parabolic equations. In this paper, we shall show the existence of global-in-time solutions to the problem. Our condition to ensure the global existence is specific to the fully…
We study the existence and uniqueness of the positive solutions of the problem (P): $\partial_tu-\Delta u+u^q=0$ ($q>1$) in $\Omega\times (0,\infty)$, $u=\infty$ on $\partial\Omega\times (0,\infty)$ and $u(.,0)\in L^1(\Omega)$, when…
In this paper, we give a clear cut relation between the the volume growth $V(r)$ and the existence of nonnegative solutions to parabolic semilinear problem \begin{align}\tag{*}\label{*} \left\{ \begin{array}{ll} \Delta u - \partial_t u +…
The purpose of this paper is to give a necessary and sufficient condition for the existence and non-existence of global solutions of the following semilinear parabolic equations \[ u_{t}=\Delta u+\psi(t)f(u),\,\,\mbox{ in }\Omega\times…
We show that all nonnegative solutions of the critical semilinear elliptic equation involving the regional fractional Laplacian are locally universally bounded. This strongly contrasts with the standard fractional Laplacian case. Second, we…
The non-existence of global solutions for semirelativistic equations with non-gauge invariant power type nonlinearity is revisited by a relatively direct way with a pointwise estimate of fractional derivative of some test functions.
We study existence and non-existence of global solutions to the semilinear heat equation with a drift term and a power-like source term, on Cartan-Hadamard manifolds. Under suitable assumptions on Ricci and sectional curvatures, we show…
It is well known from the work of [2] that the Fujita phenomenon for reaction-diffusion evolution equations with power nonlinearities does not occur on the hyperbolic space $\mathbb{H}^N$, thus marking a striking difference with the…
We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient, whose model is $$ \begin{cases} -\Delta u +…
For a given bounded domain $\Omega \subset \mathbb R^3$, with $C^2$ boundary, and a given instant of time $T>0$, we prove the existence of a global weak solution on $(0,T)$, which satisfies a maximum principle, to a parabolic $p$-Laplacian…
We study nonexistence and existence of nontrivial positive solutions to the following semilinear elliptic inequality involving gradient terms \[ \Delta u+u^p\left|\nabla u\right|^q\leq0, \] on weighted graphs, where $(p,q)\in\mathbb{R}^2$.…
In this paper we analyse the existence and non-existence of non-negative solutions to a non-local parabolic equation with a Hardy-Leray type potential. More precisely, we consider the problem $$ \begin{cases} (w_t-\Delta…
We provide sufficient and almost optimal conditions for global existence of classical solutions in parabolic H\"older spaces to quasilinear one-dimensional parabolic problems with dynamical boundary conditions.
We study semilinear elliptic equations on finite graphs with fully general exponential nonlinearities, thereby extending classical equations such as the Kazdan-Warner and Chern-Simons equations. A key contribution of this work is the…
This article investigates the existence, nonexistence, and multiplicity of positive solutions to the sublinear fractional elliptic problem $(P_{\lambda}^s)$. We begin by establishing several a priori estimates that provide regularity…