Existence results for a Cauchy-Dirichlet parabolic problem with a repulsive gradient term

Abstract

We study the existence of solutions of a nonlinear parabolic problem of Cauchy-Dirichlet type having a lower order term which depends on the gradient. The model we have in mind is the following: \[ \begin{cases}\begin{split} & u_t-\text{div}(A(t,x)\nabla u|\nabla u|^{p-2})=\gamma |\nabla u|^q+f(t,x) &\qquad\text{in } Q_T,\\ & u=0 &\qquad\text{on }(0,T)\times \partial \Omega,\\ & u(0,x)=u_0(x) &\qquad\text{in } \Omega, \end{split}\end{cases} \] where $Q_T=(0,T)\times \Omega$, $\Omega$ is a bounded domain of $\mathrm{R}^N$, $N\ge 2$, $1<p<N$, the matrix $A(t,x)$ is coercive and with measurable bounded coefficients, the r.h.s. growth rate satisfies the superlinearity condition \[ \max\left\{\frac{p}{2},\frac{p(N+1)-N}{N+2}\right\}<q<p \] and the initial datum $u_0$ is an unbounded function belonging to a suitable Lebesgue space $L^\sigma(\Omega)$. We point out that, once we have fixed $q$, there exists a link between this growth rate and exponent $\sigma=\sigma(q,N,p)$ which allows one to have (or not) an existence result. Moreover, the value of $q$ deeply influences the notion of solution we can ask for. The sublinear growth case with \[ 0<q\le\frac{p}{2} \] is dealt at the end of the paper for what concerns small value of $p$, namely $1<p<2$.