Doubly nonlinear equation involving $p(x)$-homogeneous operators: local existence, uniqueness and global behaviour

Abstract

In this work, we investigate the qualitative properties as uniqueness, regularity and stabilization of the weak solution to the nonlinear parabolic problem involving general $p(x)$-homogeneous operators: \begin{equation*} \left\{ \begin{alignedat}{2} {} \frac{q}{2q-1}\partial_t(u^{2q-1}) -\nabla.\, a(x, \nabla u) & {}= f(x,u) + h(t,x) u^{q-1} && \quad\mbox{ in } \, (0,T) \times Ω; u & {}> 0 && \quad\mbox{ in }\, (0,T) \times Ω; u & {}= 0 && \quad\mbox{ on }\, (0,T) \times \partialΩ; u(0,.)&{}= u_0 && \quad\ \mbox{in}\, \ Ω. \end{alignedat} \right. \end{equation*} Thanks to the Picone's identity obtained in [10], we prove new results about comparison principles which yield a priori estimates, positivity and uniqueness of weak solutions.