Manuel Elgueta

The blow-up problem for a semilinear parabolic equation with a potential☆

Abstract Let Ω be a bounded smooth domain in R N . We consider the problem u t = Δ u + V ( x ) u p in Ω × [ 0 , T ) , with Dirichlet boundary conditions u = 0 on ∂ Ω × [ 0 , T ) and initial datum u ( x , 0 ) = M φ ( x ) where M ⩾ 0 , φ is positive and compatible with the boundary condition. We give estimates for the blow-up time of solutions for large values of M. As a consequence of these estimates we find that, for M large, the blow-up set concentrates near the points where φ p − 1 V attains its maximum.

Uniqueness and non-uniqueness for a system of heat equations with nonlinear coupling at the boundary

We study the uniqueness problem for non negative solutions of a system of two heat equations, ut = ∆u, vt = ∆v, in a bounded smooth domain Ω, with nonlinear boundary conditions, ∂u ∂η = v, ∂v ∂η = u. We prove that for identically zero initial data, (u(x, 0), v(x, 0)) ≡ (0, 0), the zero solution is unique if and only if if pq ≥ 1. Moreover, in the case of non-negative non-trivial initial data the solution is always unique.

EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO SOME INHOMOGENEOUS NONLOCAL DIFFUSION PROBLEMS

We consider the nonlocal evolution Dirichlet problem

Analysis of an elliptic system with infinitely many solutions

u_t(x,t)=\int_{\Omega}J(\frac{x-y}{g(y)})\frac{u(y,t)}{g(y)^N}dy-u(x,t)

Asymptotic Behavior for a One-Dimensional Nonlocal Diffusion Equation in Exterior Domains

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The Asymptotic Behaviour of the Solution of a Nonlinear Diffusion Equation

x\in\Omega

Existence of generalized solutions of a nonlinear diffusion cauchy problem

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How long does it take for a gas to fill a porous container

t>0

A homotopic deformation along p of a Leray-Schauder degree result and existence for (¦u′¦p − 2u′)′ + ƒ(t, u) = 0, u(0) = u(T) = 0, p > 1☆

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How to Approximate the Heat Equation with Neumann Boundary Conditions by Nonlocal Diffusion Problems

u=0