Optimal existence, uniqueness and blow-up for a quasilinear diffusion equation with spatially inhomogeneous reaction

Abstract

Well-posedness and a number of qualitative properties for solutions to the Cauchy problem for the following nonlinear diffusion equation with a spatially inhomogeneous source∂tu=Δum+|x|σup, posed for (x,t)∈RN×(0,T), with exponents 1<p<m and σ>0, are established. More precisely, we identify the optimal class of initial conditionsu0 for which (local in time) existence is ensured and prove non-existence of solutions for the complementary set of data. We establish then (local in time) uniqueness and a comparison principle for this class of data. We furthermore prove that any non-trivial solution to the Cauchy problem blows up in a finite timeT∈(0,∞) and finite speed of propagation holds true for t∈(0,T): if u0∈L∞(RN) is an initial condition with compact support and blow-up time T>0, then u(t) is compactly supported for t∈(0,T). We also establish in this work the absence of localization at the blow-up time T for solutions stemming from compactly supported data.