Fujita type results for a quasilinear parabolic inequality of Hardy-Hénon type with time forcing terms

Abstract

We investigate a quasilinear parabolic inequality of Hardy-Hénon type with a time forcing term ut−divA(x,u,Du)≥tγ|x|σuq,x∈RN,t>0,u(x,t)≥0,x∈RN,t>0,u(x,0)=u0(x)≥0,x∈RN,where Du represents the gradient of u, |x| represents the length of x∈RN; the function A is a Carathéodory function such that A(x,z,0)=0, A(x,z,w)⋅w≥0 for every (x,z,w)∈RN×[0,∞)×RN, and on the operator A we require weak m-coercivity, namely A(x,z,w)⋅w≥c0|x|β|A(x,z,w)|mm−1,(x,z,w)∈RN×[0,∞)×RNholds for some c0>0, m≥2 , and 0≤β<Nm−1; γ>−1, σ>−N, and q>N(m−1)N−β(m−1). Based on nonlinear capacity estimates, we obtain the Fujita type results to the above problem, i.e., the conditions on the parameters γ,σ,q such that the above problem admits no nontrivial nonnegative solutions for arbitrary initial value u0.