Related papers: A nonlinear diffusion equation with reaction local…

In this work, we study the global existence of solutions for a class of semilinear nonlocal reaction-diffusion systems with $m$ components on a bounded domain $\Omega$ in $\mathbb{R}^n$ with smooth boundary. The initial data is assumed to…

Consider the nonlinear heat equation v_t-\Delta v=|v|^{p-1}v in the unit ball of R^2, with Dirichlet boundary condition. Let u_{p,K} be a radially symmetric, sign-changing stationary solution having a fixed number K of nodal regions. We…

For the quasilinear wave equation \partial_t^2u - \Delta u = u_t u_{tt}, we analyze the long-time behavior of classical solutions with small (not rotationally invariant) data. We give a complete asymptotic expansion of the lifespan and…

In this paper we consider the initial-boundary value problem for the heat, damped wave, complex-Ginzburg-Landau and Schr"odinger equations with the power type nonlinearity $|u|^p$ with $p in (1,2]$ in a two-dimensional exterior domain.…

Let $G=(V,E)$ be a locally finite, connected and weighted graph. We prove that, for a graph satisfying curvature dimension condition $CDE'(n,0)$ and uniform polynomial volume growth of degree $m$, all non-negative solutions of the equation…

We study existence, uniqueness, multiplicity and symmetry of large solutions for a class of quasi-linear elliptic equations. Furthermore, we characterize the boundary blow-up rate of solutions, including the case where the contribution of…

We consider the large-time behavior of sign-changing solutions of the inhomogeneous equation $u_t-\Delta u=|x|^\alpha |u|^{p}+\zeta(t)\,{\mathbf w}(x)$ in $(0,\infty)\times\mathbb{R}^N$, where $N\geq 3$, $p>1$, $\alpha>-2$, $\z, {\mathbf…

We consider the blow-up of solutions for a semilinear reaction diffusion equation with exponential reaction term. It is know that certain solutions that can be continued beyond the blow-up time possess a nonconstant selfsimilar blow-up…

We study the self-similar blow-up profiles associated to the following second order reaction-diffusion equation with strong weighted reaction and unbounded weight: $$ \partial_tu=\partial_{xx}(u^m) + |x|^{\sigma}u^p, $$ posed for…

We classify the finite time blow-up profiles for the following reaction-diffusion equation with unbounded weight: $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed in any space dimension $x\in\mathbf{R}^N$, $t\geq0$ and with exponents…

In this paper, we show global existence and non-existence results for the heat equation with some of the squares of smooth vector fields on $\Rn$ satisfying H\"{o}rmander's rank condition with a non-linearity of the form $f(u)$, where $f$…

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,…

We prove existence and uniqueness of \emph{eternal solutions} in self-similar form growing up in time with exponential rate for the weighted reaction-diffusion equation $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed in $\real^N$, with…

We study the blow-up problem of one-dimensional nonlinear heat equations. Our result shows that for a certain class of initial conditions, the solutions blow up in finite time and we characterize the asymptotic dynamics of these solutions.…

We consider the problem of existence of a solution $u$ to $\partial_t u-\partial_{xx} u = 0$ in $(0,T)\times\mathbb{R}_+$ subject to the boundary condition $-u_x(t,0)+g(u(t,0))=\mu$ on $(0,T)$ where $\mu$ is a measure on $(0,T)$ and $g$ a…

We give a sufficient condition for non-existence of global nonnegative mild solutions of the Cauchy problem for the semilinear heat equation $u' = Lu + f(u)$ in $L^p(X,m)$ for $p \in [1,\infty)$, where $(X,m)$ is a $\sigma$-finite measure…

We consider the semilinear heat equation $$u_t-\Delta u=f(u) $$ for a large class of non scale invariant nonlinearities of the form $f(u)=u^pL(u)$, where $p>1$ is Sobolev subcritical and $L$ is a slowly varying function (which includes for…

This paper is concerned with the existence/nonexistence of nontrivial global-in-time solutions to the Cauchy problem \begin{equation} \begin{cases}\tag{P}\partial_tu-\partial_x^2u+Vu=(1+x^2)^{-\frac{m}{2}}u^p,&x\in\mathbb{R},\ t>0,\\…

We study positive radial solutions of quasilinear elliptic systems with a gradient term in the form $$ \left\{ \begin{aligned} \Delta_{p} u&=v^{m}|\nabla u|^{\alpha}&&\quad\mbox{ in }\Omega,\\ \Delta_{p} v&=v^{\beta}|\nabla u|^{q}…

In this paper, we study the blow-up of solutions to the semilinear Moore-Gibson-Thompson (MGT) equation with nonlinearity of derivative type $|u_t|^p$ in the conservative case. We apply an iteration method in order to study both the…