On the lifespan of classical solutions to a non-local porous medium problem with nonlinear boundary conditions

2019/04/24 by Marras, Monica, Pintus, Nicola, Viglialoro, Giuseppe · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1904.10747

Abstract

In this paper we analyze the porous medium equation %\begincases u_t=Δu^m + a\io u^p-b u^q -c|∇√(u)|² \textrmin Ω× I,%
%u_ν-g(u)=0 amp; \textrmon ∂ Ω, tgt;0,
%u(\bf x,0)=u₀(\bf x)amp;\bf x ∈ Ω,
%\endcases where Ω is a bounded and smooth domain of \R^N, with N≥ 1, and I= [0,t^*) is the maximal interval of existence for u. The constants a,b,c are positive, m,p,q proper real numbers larger than 1 and the equation is complemented with nonlinear boundary conditions involving the outward normal derivative of u. Under some hypothesis on the data, including intrinsic relations between m,p and q, and assuming that for some positive and sufficiently regular function u₀(\nx) the Initial Boundary Value Problem (IBVP) associated to \eqrefProblemAbstract possesses a positive classical solution u=u(\nx,t) on Ω× I: \beginitemize \item [\triangleright] when p>q and in 2- and 3-dimensional domains, we determine a lower bound of t^* for those u becoming unbounded in L^m(p-1)(Ω) at such t^*; \item [\triangleright] when p

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