Raúl Ferreira

Critical exponents for a semilinear parabolic equation with variable reaction

In this paper we study the blow-up phenomenon for nonnegative solutions to the following parabolic problem: ut(x, t) = ∆u(x, t) + (u(x, t)) , in Ω× (0, T ), where 0 1. When Ω = R we show that if p− > 1 + 2/N then there are global nontrivial solutions while if 1 < p− ≤ p+ ≤ 1+2/N then all solutions to the problem blow up in finite time. Moreover, in case p− < 1+2/N < p+ there are functions p(x) such that all solutions blow up in finite time and functions p(x) such that the problem possesses global nontrivial solutions. When Ω is a bounded domain we prove that there are functions p(x) and domains Ω such that all solutions to the problem blow up in finite time. On the other hand, if Ω is small enough then the problem possesses global nontrivial solutions regardless the size of p(x).

Uniform Bounds for the Best Sobolev Trace Constant

Abstract We study the Sobolev trace embedding W1,p(Ω) ↪ Lq(∂Ω), looking at the dependence of the best constant and the extremals on p and q. We prove that there exists a uniform bound (independent of (p, q)) for the best constant if and only if (p, q) lies far from (N, ∞). Also we study some limit cases, q = ∞ with p > N or p = ∞ with 1 ≤ q ≤ ∞.

Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models

. Let J : R → R be a nonnegative, smooth function with ∫R J(r)dr = 1, supported in [-1,1], symmetric, J(r) = J(-r), and strictly increasing in [-1,0]. We consider the Neumann boundary value problem for a nonlocal, nonlinear operator that is similar to the porous medium, and we study the equation U t (x,t)= ∫L-L (J(x-y u(y,t))-J(x-y u(x,t)))dy x∈ [-L,L]. We prove existence and uniqueness of solutions and a comparison principle. We find the asymptotic behaviour of the solutions as t →∞: they converge to the mean value of the initial data. Next, we consider a discrete version of the above problem. Under suitable hypotheses we prove that the discrete model has properties analogous to the continuous one. Moreover, solutions of the discrete problem converge to the continuous ones when the mesh parameter goes to zero. Finally, we perform some numerical experiments.

The Interfaces of an Inhomogeneous Porous Medium Equation with Convection

ABSTRACT We study the evolution of the interfaces of solutions to the nonhomogeneous porous medium equation with convection with n, m > 1, and ρ(x) > 0, ρ(x) → 0 as |x|→∞. We characterize when the interfaces go to ±infin; in finite time in terms of ρ. Since the equation is not symmetric, the left and right interfaces behave in a completely different way. Moreover, their behaviour strongly departs from the purely diffusive case.

Grow-up for a quasilinear heat equation with a localized reaction in higher dimensions

We study the behaviour of nonnegative global solutions to the quasilinear heat equation with a reaction localized in a ball

On the quenching set for a fast diffusion equation: regional quenching

\begin{aligned} u_t={\varDelta } u^m+a(x)u^p, \end{aligned}

Extinction behavior for fast diffusion equations with absorption

ut=Δum+a(x)up,for