相关论文: Existence of Nonnegative Solutions of Nonlinear Fr…

This paper aims to study the asymptotic behaviour of the fundamental solutions (heat kernels) of non-local (partial and pseudo differential) equations with fractional operators in time and space. In particular, we obtain exact asymptotic…

Let $G=(V,E)$ be a locally finite connected weighted graph, and $\Omega$ be an unbounded subset of $V$. Using Rothe's method, we study the existence of solutions for the semilinear heat equation $\partial_tu+|u|^{p-1}\cdot u=\Delta…

In this paper, we are concerned with the following type of fractional problems: $$ \begin{cases}\dis (-\Delta)^{s} u-\mu\frac{u}{|x|^{2s}}-\lambda u=|u|^{2^*_{s}-2}u+f(x,u), &\text{in} \Omega,\ \ \, u=0\,&\text{in} \R^N\backslash\Omega…

We establish the existence and uniqueness of fundamental solutions for the fractional porous medium equation introduced in \cite{PQRV1}. They are self-similar functions of the form $u(x,t)= t^{-\alpha} f(|x|\,t^{-\beta})$ with suitable…

We establish quantitative estimates for solutions $u(t,x)$ to the fractional nonlinear diffusion equation, $\partial_t u +(-\Delta)^s (u^m)=0$ in the whole range of exponents $m>0$, $0<s<1$. The equation is posed in the whole space…

In this paper we investigate the nonexistence of nonnegative solutions of parabolic inequalities of the form $$\begin{cases} &u_t \pm L_\mathcal A u\geq (K\ast u^p)u^q \quad\mbox{ in } \mathbb R^N \times \mathbb (0,\infty),\, N\geq 1,\\…

In this article the problem to be studied is the following $$ (P) \left\{ \begin{array}{rcll} u_t+(-\D^s_{p}) u & = & f(x,t) & \text{ in } \O_{T}\equiv \Omega \times (0,T), \\ u & = & 0 & \text{ in }(\ren\setminus\O) \times (0,T), \\ u &…

We study the regularity properties of the solutions to the nonlinear equation with fractional diffusion $$ \partial_tu+(-\Delta)^{\sigma/2}\varphi(u)=0, $$ posed for $x\in \mathbb{R}^N$, $t>0$, with $0<\sigma<2$, $N\ge1$. If the…

We consider time fractional stochastic heat type equation $$\partial^\beta_tu(t,x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\sigma(u)\stackrel{\cdot}{W}(t,x)]$$ in $(d+1)$ dimensions, where $\nu>0$, $\beta\in (0,1)$, $\alpha\in (0,2]$,…

This work concerns with the existence of solutions for the following class of nonlocal elliptic problems \begin{equation*}\label{00} \left\{ \begin{array}{l} (-\Delta)^{s}u + u = |u|^{p-2}u\;\;\mbox{in $\Omega$},\\ u \geq 0 \quad \mbox{in}…

In this article, we consider the space-time fractional (nonlocal) equation characterizing the so-called "double-scale" anomalous diffusion $$\partial_t^\beta u(t, x) = -(-\Delta)^{\alpha/2}u(t,x) - (-\Delta)^{\gamma/2}u(t,x) \ \ t> 0, \…

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…

In this paper, we study the positive solutions to the following singular and non local elliptic problem posed in a bounded and smooth domain $\Omega\subset \R^N$, $N> 2s$: % \begin{eqnarray*} (P_\lambda)\left\{\begin{array}{lll}…

In this paper, we consider the fractional heat equation $u_{t}=\triangle^{\alpha/2}u+f(u)$ with Dirichlet boundary conditions on the ball $B_{R}\subset \mathbb{R}^{d}$, where $\triangle^{\alpha/2}$ is the fractional Laplacian,…

We consider nonnegative solutions of the quasilinear heat equation $\partial_t u = \tfrac{1}{2} u \partial_x^2 u$ in one dimension. Our solutions may vanish and may be unbounded. The equation is then degenerate, and weak solutions are…

We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularities \begin{align} (-\Delta)_{p(\cdot)}^{s}…

We investigate the existence and nonexistence of positive solutions for the quasilinear elliptic inequality $L_\mathcal{A} u= -{\rm div}[\mathcal{A}(x, u, \nabla u)]\geq (I_\alpha\ast u^p)u^q$ in $\Omega$, where $\Omega\subset \mathbb{R}^N,…

We proceed with the investigation of the problem $(P_\lambda): $ $-\Delta u = \lambda b(x)|u|^{q-2}u +a(x)|u|^{p-2}u \ \mbox{ in } \Omega, \ \ \frac{\partial u}{\partial \mathbf{n}} = 0 \ \mbox{ on } \partial \Omega$, where $\Omega$ is a…

Our purpose of this paper is to study the nonexistence of nonnegative very weak solutions of \begin{equation}\label{eq 0.1} \displaystyle (-\Delta)^\alpha u = u^p+\nu\quad {\rm in}\quad \Omega,\qquad\ u=g\quad {\rm in}\quad \mathbb{…

This paper concerns the global in time existence of solutions for a semilinear heat equation \begin{equation} \tag{P} \label{eq:P} \begin{cases} \partial_t u = \Delta u + f(u), &x\in \mathbb{R}^N, \,\,\, t>0, \\[3pt] u(x,0) = u_0(x) \ge 0,…