Blow-up and lifespan of solutions to a nonlocal parabolic equation at arbitrary initial energy level

We consider a nonlocal parabolic equation. By exploiting the boundary condition and the variational structure of the equation, we prove finite time blow-up of the solution for initial data at arbitrary energy level. We also obtain the lifespan of the blow-up solution. The results generalize the former studies on this equation.     

Blow-up for a degenerate parabolic equation with a nonlocal source

In this paper, we investigate the blowup properties of the positive solutions to the following nonlocal degenerate parabolic equation

     

Blow-up for a degenerate parabolic equation with nonlocal source and nonlocal boundary

In this article, we investigate the blow-up properties of the positive solutions for a doubly degenerate parabolic equation with nonlocal source and nonlocal boundary condition. The conditions on the existence and nonexistence of global positive solutions are given. Moreover, we give the precise blow-up rate estimate and the uniform blow-up estimate for the blow-up solution.     

Blow-up phenomena for a doubly degenerate equation with positive initial energy

In this paper, an initial boundary value problem related to the equation

     

Blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions

In this paper we give a positive answer to the conjecture proposed in [A. El Soufi, M. Jazar, R. Monneau, A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (1) (2007) 17–39] by El Soufi et al. concerning the finite time blow-up for solutions of the problem (1), (2) below. More precisely, we give a direct proof of [A. El Soufi, M. Jazar, R. Monneau, A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (1) (2007) 17–39, Theorem 1.1] and the conjecture given for the case p>2$p>2$.     

On blow-up of solutions for a semilinear parabolic equation involving variable source and positive initial energy

This article is concerned with a class of semilinear parabolic equations with variable reaction $u_t=\Delta u+u^{p\left(x\right)}$ with homogeneous Dirichlet boundary conditions. Under some appropriate assumptions on the parameters, and with certain initial data, a blow-up result is established with positive initial energy.     

Blow-up problem for semilinear heat equation with absorption and a nonlocal boundary condition

In this paper we consider a semilinear parabolic equation u_t=Δu−c(x,t)u^p for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition u∣_{∂Ω×(0,∞)}=∫_Ωk(x,y,t)u^ldy and nonnegative initial data where p>0 and l>0. We prove some global existence results. Criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data are also given.     

Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition

In this paper, we consider a semilinear heat equation ut=Δu+c(x,t)up for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition and nonnegative initial data where p>0 and l>0. We prove global existence theorem for max(p,l)?1. Some criteria on this problem which determine whether the solutions blow up in a finite time for sufficiently large or for all nontrivial initial data or the solutions exist for all time with sufficiently small or with any initial data are also given.     

带非局部源的退化半线性抛物方程的解的爆破性质

This paper deals with the blow-up properties of the positive solutions to the nonlocal degenerate semilinear parabolic equation u _t − (x ^a u _x ) _x =∫ ₀ ^a f(u)dx in (0,a) × (0,T) under homogeneous Dirichlet conditions. The local existence and uniqueness of classical solution are established. Under appropriate hypotheses, the global existence and blow-up in finite time of positve solutions are obtained. It is also proved that the blow-up set is almost the whole domain. This differs from the local case. Furthermore, the blow-up rate is precisely determined for the special case: f(u)=u ^p , p>1.     

A numerical approach for a semilinear parabolic equation with a nonlocal boundary condition

A semilinear reaction-diffusion problem with a nonlocal boundary condition is studied. This paper presents a new and very easy implementable numerical algorithm for computations. This is based on a suitable linearization in time and on the principle of linear superposition. Any method for the space discretization (FEM was taken in this analysis) can be chosen. The derived algorithm is implicit and it does not need any iteration scheme to get a solution with the nonlocal boundary condition. Stability analysis has been performed and the optimal error estimates have been derived. Numerical results have been compared with other known techniques.     

Blow-up profiles for positive solutions of nonlocal dispersal equation

In this paper, we study the blow-up profiles of the nonlocal dispersal equation. More precisely, we prove that the positive solution of nonlocal dispersal equation has different blow-up profiles, depending on the refuge domain.     

A singular solution with smooth initial data for a semilinear parabolic equation

We consider the Cauchy problem for a parabolic partial differential equation with a power nonlinearity. Our concern in this paper is the existence of a singular solution with smooth initial data. By using the Haraux-Weissler equation, it is shown that there exist singular forward self-similar solutions. Using this result, we also obtain a sufficient condition for the singular solution with general initial data including smooth initial data.     

Blow-up of solutions of a semilinear hyperbolic equation and a parabolic equation with general forcing term and boundary condition

Semilinear hyperbolic and parabolic initial–boundary value problems are studied. Criteria for solutions of a semilinear hyperbolic equation and a parabolic equation with general forcing term and general boundary condition to blow up in finite time are obtained.     

Existence of solutions for a higher-order semilinear parabolic equation with singular initial data

We establish the existence of solutions of the Cauchy problem for a higher-order semilinear parabolic equation by introducing a new majorizing kernel. We also study necessary conditions on the initial data for the existence of local-in-time solutions and identify the strongest singularity of the initial data for the solvability of the Cauchy problem.