Parabolic systems with \({p,q}\)-growth: a variational approach (Q394032)

Let \(\Omega\subset\mathbb R^n\) (\(n\geq 2\)) be a bounded open set with boundary \(\partial\Omega\) and let \(T>0\) be a fixed real number. The authors consider the Cauchy-Dirichlet problem \[ \begin{cases} \partial_t u-\text{div}\;Df(Du)=0\;\; &\text{ in }\; \Omega_T=\Omega\times(0,T),\\ u=g\;\;&\text{ on }\;\partial_{\mathcal P}\Omega_T, \end{cases} \] where \(u:\;\Omega_T\subset\mathbb R^{n+1}\to\mathbb R^N\), with \(n\geq 2\) and \(N\geq 1\), is a vector-valued function with values in \(\mathbb R^N\) and where \(\partial_{\mathcal P}\Omega_T:=[\partial\Omega\times(0,T)]\cup[\bar\Omega\times\{0\}]\) denotes the parabolic boundary of \(\Omega_T\) and \(f:\;\mathbb R^{Nn}\to\mathbb R_+\) is an integrand of class \(C^1\) satisfying some non-standard \(p,q\)-growth assumption and some monotonicity conditions. The notions of variational and weak solutions are defined. Some existence, uniqueness and regularity results of variational and weak solutions are obtained.