Related papers: Nonlinear Convection in Reaction-diffusion Equatio…
We discuss the H\'{e}non parabolic equation $\partial_t u = \Delta u + |x|^\sigma u^p$ in a finite ball in $\mathbb{R}^N$ under the Dirichlet boundary condition, where $N\ge1$, $p>1$, and $\sigma>0$. We assume that the exponent $p$ is…
In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation \[ u_t-\mathrm{div}\left(|\nabla u|^{p-2}\nabla u\right)=-|u|^{\beta-1}u+\alpha|u|^{q-2}u, \] where $p>1,\beta>0$, $q\geq1$ and $\alpha>0$.…
We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. In some earlier works [1, 2], we constructed a solution $u$ for that equation such that $u$ and…
We study the behaviour of the solutions to the quasilinear heat equation with a reaction restricted to a half-line $$ u_t=(u^m)_{xx}+a(x) u^p, $$ $m, p>0$ and $a(x)=1$ for $x>0$, $a(x)=0$ for $x<0$. We first characterize the global…
In this study, we examine a double nonlinear porous medium equation subject to a novel nonlinearity condition within a bounded domain. First, we introduce the blow-up solution for the problem under consideration for the negative initial…
The finite time blow-up of solutions for 1-D NLS with oscillating nonlinearities is shown in two domains: (1) the whole real line where the nonlinear source is acting in the interior of the domain and (2) the right half-line where the…
We investigate finite-time blow-up for nonnegative solutions to the Cauchy problem associated with semilinear parabolic equations driven by a mixed local--nonlocal operator. The reaction term is assumed to satisfy suitable structural…
The Neumann problem in balls $\Omega\subset\mathbb{R}^n$, $n\in\{3,4\}$, for the chemotaxis system \begin{equation*} \left\{ \begin{array}{ll} u_t = \Delta u - \nabla \cdot (u\nabla v), \\[1mm] 0 = \Delta v - \mu^{(w)}(t) + w, \quad…
The authors of this paper study singular phenomena(vanishing and blowing-up in finite time) of solutions to the homogeneous $\hbox{Dirichlet}$ boundary value problem of nonlinear diffusion equations involving $p(x)$-\hbox{Laplacian}…
This paper deals with the following Petrovsky equation with damping and nonlinear source \[u_{tt}+\Delta^2 u-M(\|\nabla u\|_2^2)\Delta u-\Delta u_t+|u_t|^{m(x)-2}u_t=|u|^{p(x)-2}u\] under initial-boundary value conditions, where $M(s)=a+…
This paper studies a nonlinear plate equation with internal fractional damping and a time-delay term, driven by a polynomial-type nonlinear source. Such a model arises naturally in the description of viscoelastic and feedback-controlled…
We investigate pointwise upper bounds for nonnegative solutions $u(x,t)$ of the nonlinear initial value problem \begin{equation}\label{0.1} 0\leq(\partial_t-\Delta)^\alpha u\leq u^\lambda \quad\text{ in }\mathbb{R}^n…
We extend the slow blow up solutions of Krieger, Schlag, and Tataru to semilinear wave equations on a curved background. In particular, for a class of manifolds $(M,g)$ we show the existence of a family of blow-up solutions with finite…
We consider the stochastic heat equation with multiplicative white noise: $\partial_t u =\partial_x^2u + b(u) +\sigma(u) \dot W$, both on $[0,1]$ and $\mathbf{R}$. In the case of $[0,1]$ we show that the finite Osgood criterion on $b$ is a…
The first part of this paper is devoted to the derivation of a technical result, related to the stability of the solution of a reaction-diffusion equation $u_t-\Delta u = f(x,u)$ on $(0,\infty)\times \mathbb{R}^N$, where the initial datum…
In this paper we are interested in propagation phenomena for nonlocal reaction-diffusion equations of the type: $\delta_tu = J \times u - u + f (x, u) t \in R^+, x \in R^N$, where J is a probability density and f is a KPP nonlinearity…
Second initial boundary problem in narrow domains of width $\epsilon\ll 1$ for linear second order differential equations with nonlinear boundary conditions is considered in this paper. Using probabilistic methods we show that the solution…
This paper investigates the repulsive chemotaxis-consumption model \begin{align*} \partial_t u &= \nabla \cdot (D(u) \nabla u) + \nabla \cdot (u \nabla v), \\ 0 &= \Delta v - uv \end{align*} in an $n$-dimensional ball, $n \ge 3$, where the…
This paper is concerned with the blowup phenomenon of stochastic parabolic equations both on bounded domain and in the whole space. We introduce a new method to study the blowup phenomenon on bounded domain. Comparing with the existing…
We consider the scaling critical Lebesgue norm of blow-up solutions to the semilinear heat equation $u_t=\Delta u+|u|^{p-1}u$ in an arbitrary smooth domain of $\mathbf{R}^n$. In the range $p>p_S:=(n+2)/(n-2)$, we show that the critical norm…