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We consider evolutionary problems associated with a convex integrand $${f : \Omega_T \times {\mathbb R}^{Nn} \to [0, \infty)}$$f:ΩT×RNn→[0,∞), which is $${\alpha}$$α-Hölder continuous with respect to the x-variable and satisfies a non-standard p, q-growth condition. We prove the existence of weak solutions $${u : \Omega_T \to {\mathbb R}^N}$$u:ΩT→RN, which solve$$\partial_{t} u - {\rm div} \partial_{\zeta} f(x, t, Du) = 0$$∂tu-div∂ζf(x,t,Du)=0weakly in $${\Omega_{T}}$$ΩT. Therefore, we use the concept of variational solutions, which exist under a mild assumption on the gap q − p, namely$$\frac{2n}{n+2} < p \leq q < p + 1.$$2nn+2<p≤q<p+1.For$$\frac{2n}{n+2} < p \leq q < p + \frac{{\rm min}\{2, p\} \alpha}{n+2},$$2nn+2<p≤q<p+min{2,p}αn+2,we prove that the spatial derivative Du of a variational solution u admits a higher integrability and is accordingly a weak solution.

Existence of weak solutions of parabolic systems with p, q-growth