arXiv subjects
Publications and source records attributed to Raul Ferreira.
We study the possibility of non-simultaneous blow-up for positive solutions of a coupled system of two semilinear equations, $u_t = J*u-u+ u^αv^p$, $v_t =Δv^+u^qv^β$, $p, q, α, β>0$ with homogeneous Dirichlet boundary conditions and positive initial data. We also give the blow-up rates for non-simultaneous blow-up.
In this paper we study the Dirichlet eigenvalue problem $$ -Δ_p u-Δ_{J,p}u =λ|u|^{p-2}u \quad \text{ in } Ω,\quad u=0 \quad\text{ in } Ω^c=\mathbb{R}^N\setminusΩ. $$ Here $Δ_p u$ is the standard local $p-$Laplacian, $Δ_{J,p}u$ is a nonlocal, $p-$homogeneous operator of order zero and $Ω$ is a bounded domain in $\mathbb{R}^N$. We show that the first eigenvalue (that is isolated and simple) satisfies $(λ_1)^{1/p}\to Λ$ as $p\to\infty$ where $Λ$ can be characterized in terms of the geometry of $Ω$. We also find that the eigenfunctions converge, $u_\infty=\lim_{p\to\infty} u_p$, and find the limit problem that is satisfied in the limit.
We study the behaviour of global solutions to the quasilinear heat equation with a reaction localized $$ u_t=(u^m)_{xx}+a(x) u^p, $$ $m, p>0$ and $a(x)$ being the characteristic function of an interval. we prove that there exists $p_0=\max\{1,\frac{m+1}2\}$ such that all global solution are bounded if $p>p_0$, while for $p\le p_0$ all the solution are global and unbounded. In the last case, we prove that if $p m$ the grow-up rate coincides with that rate, but only inside the support of $a$; outside the interval the rate is smaller.
We study the behaviour of nonnegative solutions to the quasilinear heat equation with a reaction localized in a ball $$ u_t=Δu^m+a(x)u^p, $$ for $m>0$, $0<p\le\max\{1,m\}$, $a(x)=\mathds{1}_{B_L}(x)$, $0<L<\infty$ and $N\ge2$. We study when solutions, which are global in time, are bounded or unbounded. In particular we show that the precise value of the length $L$ plays a crucial role in the critical case $p=m$ for $N\ge3$. We also obtain the asymptotic behaviour of unbounded solutions and prove that the grow-up rate is different in most of the cases to the one obtained when $L=\infty$.
n this paper we study the asymptotic behavior for a nonlocal heat equation in an inhomogenous medium: $$ρ(x)u_t=J\ast u-u \text{in}\mathbb{R}^N\times (0,\infty)\,,$$ where $ρ$ is a continous positive function, $u$ is nonnegative and $J$ is a probability measure having finite second-order momentum. Depending on integrability conditions on the initial data $u_0$ and $ρ$, we prove various isothermalisation results, i.e. $u(t)$ converges to a constant state in the whole space.
We consider the Cauchy problem posed in the whole space for the following nonlocal heat equation: u_t = J * u - u, where J is a symmetric continuous probability density. Depending on the tail of J, we give a rather complete picture of the problem in optimal classes of data by: (i) estimating the initial trace of (possibly unbounded) solutions; (ii) showing existence and uniqueness results in a suitable class; (iii) giving explicit unbounded polynomial solutions.