Carmen Cortázar

Symmetry in an elliptic problem and the blow—up set of

We will consider in this paper a semilinear elliptic equation {triangle}u + f(u) = 0 in {Omega}, (1.5) where the function f is locally Lipschitz in (0,{infinity}) and continuous in (0,{infinity}). We study symmetry properties of nonnegative solutions of this equation in two different situations: first we assume {Omega} = IR{sup N}, and second we consider {Omega} {ne} IR{sup N} and we provide (1.5) with overdetermined boundary conditions. Next we describe our results in the first case, that is, when {Omega} = IR{sup N}. We will consider the following hypothesis on the nonlinear function f (F) f(0) {le} 0, f continuous in (0,+{infinity}), locally Lipschitz in (0,+{infinity}) and there exists {alpha} > 0 so that f is strictly decreasing in [0,{alpha}]. We note that the support of a solution of (1.5) is not known a priori and so we have in fact a free boundary involved. Our goal is to determine the shape of this support and the symmetry properties of the solution.

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.

ON THE UNIQUENESS OF SIGN CHANGING BOUND STATE SOLUTIONS OF A SEMILINEAR EQUATION

Abstract We establish the uniqueness of the higher radial bound state solutions of (P) Δ u + f ( u ) = 0 , x ∈ R n . We assume that the nonlinearity f ∈ C ( − ∞ , ∞ ) is an odd function satisfying some convexity and growth conditions, and has one zero at b > 0 , is nonpositive and not-identically 0 in ( 0 , b ) , positive in [ b , ∞ ) , and is differentiable in ( 0 , ∞ ) .

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

The application of dissipative operators to nonlinear diffusion equations

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

Analysis of an elliptic system with infinitely many solutions

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

x\in\Omega

On the Uniqueness of the Limit for an Asymptotically Autonomous Semilinear Equation on ℝ N

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The Asymptotic Behaviour of the Solution of a Nonlinear Diffusion Equation

t>0