Existence of renormalized solutions for parabolic equations without the sign condition and with three unbounded nonlinearities
Volume 39 / 2012
Applicationes Mathematicae 39 (2012), 1-22
MSC: Primary 47A15; Secondary 46A32, 47D20.
DOI: 10.4064/am39-1-1
Abstract
We study the problem \begin{gather*} \frac{\partial b(x, u)}{\partial t}-{\mathop{\rm div}}(a(x,t,u,D u))+H(x,t,u,Du) = \mu\quad \text{in } Q=\varOmega\times (0,T),\\ b(x,u)|_{t=0}=b(x,u_0)\quad\text{in } \varOmega,\\ u=0\quad \text{in } \partial\varOmega\times (0,T). \end{gather*} The main contribution of our work is to prove the existence of a renormalized solution without the sign condition or the coercivity condition on $H(x,t,u,Du)$. The critical growth condition on $H$ is only with respect to $Du$ and not with respect to $u$. The datum $\mu$ is assumed to be in $L^1(Q)+L^{p'}(0,T;W^{-1, p'}(\varOmega))$ and $b(x,u_0)\in L^1(\varOmega)$.
