Parabolic Systems with p, q-Growth: A Variational Approach

Abstract

We consider the evolution problem associated with a convex integrand \({f : \mathbb{R}^{Nn}\to [0,\infty)}\) satisfying a non-standard p, q-growth assumption. To establish the existence of solutions we introduce the concept of variational solutions. In contrast to weak solutions, that is, mappings \({u\colon \Omega_T \to \mathbb{R}^n}\) which solve $$ \partial_tu-{\rm div} Df(Du)=0 $$ weakly in \({\Omega_T}\), variational solutions exist under a much weaker assumption on the gap q − p. Here, we prove the existence of variational solutions provided the integrand f is strictly convex and $$\frac{2n}{n+2} < p \le q < p+1.$$ These variational solutions turn out to be unique under certain mild additional assumptions on the data. Moreover, if the gap satisfies the natural stronger assumption $$ 2\le p \le q < p+ {\rm min}\big \{1,\frac{4}{n} \big \},$$ we show that variational solutions are actually weak solutions. This means that solutions u admit the necessary higher integrability of the spatial derivative Du to satisfy the parabolic system in the weak sense, that is, we prove that $$u\in L^q_{\rm loc}\big(0,T; W^{1,q}_{\rm loc}(\Omega,\mathbb{R}^N)\big).$$