The effect of Harnack inequality on the existence and nonexistence results for quasi-linear parabolic equations related to Caffarelli-Kohn-Nirenberg inequalities

摘要

In this work we study the problem 1 $$ left{ {begin{array}{lll} {u_t-{rm div}(|x|^{-p gamma} |nabla u|^{p-2}nabla u) = lambdafrac{u^{alpha}}{|x|^{p(gamma + 1)}}+f,{rm in}, Omega times (0,T),} {u geq 0,{rm in},Omega times (0,T),{rm and},u = 0,{rm on}, partialOmega ,times (0,T),} {u(x,0) = u_{0}(x),{rm in},Omega,} end{array} } right. $$ where $Omega subset {it mathbb R}^{N} (N geq 3)$ is a bounded regular domain such that $0 in Omega$ , $alpha geq p-1, -infty < gamma < frac{N-p}{p},lambda > 0, f in L^{1}(Omega times (0, T))$ and $u_{0} in L^{1}(Omega)$ are positive functions. The main points under analysis are some nonexistence results and complete blow-up in the case $p > 2$ and $gamma + 1 > 0$ and some examples of existence for $(gamma +1) > 0$ and $1 < p < 2$ . These results are interesting as they prove the role of Harnack inequality in this kind of problems and allow to understand better the blow-up behavior.