Existence and nonexistence results for a class of linear and semilinear parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities

Abdellaoui, Boumediene, Colorado, Eduardo, and Peral, Ireneo. "Existence and nonexistence results for a class of linear and semilinear parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities." Journal of the European Mathematical Society 006.1 (2004): 119-148. <http://eudml.org/doc/277448>.

@article{Abdellaoui2004,
abstract = {In this work we study the problem $u_t−\operatorname\{div\}(|x|^\{−2\gamma \}\nabla u)=\lambda \frac\{u^\alpha \}\{|x|^\{2(\gamma +1)\}\}+f$ in $\Omega \times (0,T)$, $u\ge 0$ in $\Omega \times (0,T)$, $u=0$ on $\partial \Omega \times (0,T)$, $u(x,0)=u_0(x)$ in $\Omega $, $\Omega \subset \mathbb \{R\}^N$$(N\ge 2)$ is a bounded regular domain such that $0\in \Omega $, $\lambda >0$, $\alpha >0$, $-\infty <\gamma <(N−2)/2$, $f$ and $u_0$ are positive functions such that $f\in L^1(\Omega \times (0,T))$ and $u_0\in L^1(\Omega )$. The main points under analysis are: (i) spectral instantaneous and complete blow-up related to the Harnack inequality in the case $\alpha =1$, $1+\gamma >0$; (ii) the nonexistence of solutions if $\alpha >1$, $1+\gamma >0$; (iii) a uniqueness result for weak solutions (in the distribution sense); (iv) further results on existence of weak solutions in the case $0<\alpha \le 1$.},
author = {Abdellaoui, Boumediene, Colorado, Eduardo, Peral, Ireneo},
journal = {Journal of the European Mathematical Society},
keywords = {Harnack inequality; Caffarelli-Kohn-Nirenberg inequalities; complete and instantaneous blow-up for solution of parabolic equations; weak solutions; entropy solutions; spectral instantaneous and complete blow-up; Harnack inequality; spectral instantaneous and complete blow-up; Harnack inequality},
language = {eng},
number = {1},
pages = {119-148},
publisher = {European Mathematical Society Publishing House},
title = {Existence and nonexistence results for a class of linear and semilinear parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities},
url = {http://eudml.org/doc/277448},
volume = {006},
year = {2004},
}

TY - JOUR
AU - Abdellaoui, Boumediene
AU - Colorado, Eduardo
AU - Peral, Ireneo
TI - Existence and nonexistence results for a class of linear and semilinear parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities
JO - Journal of the European Mathematical Society
PY - 2004
PB - European Mathematical Society Publishing House
VL - 006
IS - 1
SP - 119
EP - 148
AB - In this work we study the problem $u_t−\operatorname{div}(|x|^{−2\gamma }\nabla u)=\lambda \frac{u^\alpha }{|x|^{2(\gamma +1)}}+f$ in $\Omega \times (0,T)$, $u\ge 0$ in $\Omega \times (0,T)$, $u=0$ on $\partial \Omega \times (0,T)$, $u(x,0)=u_0(x)$ in $\Omega $, $\Omega \subset \mathbb {R}^N$$(N\ge 2)$ is a bounded regular domain such that $0\in \Omega $, $\lambda >0$, $\alpha >0$, $-\infty <\gamma <(N−2)/2$, $f$ and $u_0$ are positive functions such that $f\in L^1(\Omega \times (0,T))$ and $u_0\in L^1(\Omega )$. The main points under analysis are: (i) spectral instantaneous and complete blow-up related to the Harnack inequality in the case $\alpha =1$, $1+\gamma >0$; (ii) the nonexistence of solutions if $\alpha >1$, $1+\gamma >0$; (iii) a uniqueness result for weak solutions (in the distribution sense); (iv) further results on existence of weak solutions in the case $0<\alpha \le 1$.
LA - eng
KW - Harnack inequality; Caffarelli-Kohn-Nirenberg inequalities; complete and instantaneous blow-up for solution of parabolic equations; weak solutions; entropy solutions; spectral instantaneous and complete blow-up; Harnack inequality; spectral instantaneous and complete blow-up; Harnack inequality
UR - http://eudml.org/doc/277448
ER -