Existence of solutions to an initial Dirichlet problem of evolutional \(p(x)\)-Laplace equations (Q432266)

The authors study the existence and uniqueness of weak solutions of the following initial-boundary problem NEWLINE\[NEWLINE \begin{aligned} & u_t = \text{div}\left(|\nabla u|^{p(x)-2}\nabla u\right) + f(x,t,u)\\ & \left.u(x,0)\right|_{\Gamma_T} 0, \quad \left.u\right|_{t = 0} = u_0(x), \end{aligned} NEWLINE\]NEWLINE where \(\Omega\) is a domain of \(\mathbb{R}^N\) with Lipschitz continuous boundary \(\partial\Omega\), \(\Gamma_T = \partial\Omega \times [0,T]\) and \(p(x)\) is a measurable function with \(\inf p(x) > 2\). This problem describe the motion of generalized Newtonian fluids.NEWLINENEWLINEIn the study of the global existence of solutions, the following hypothesis to the function \(f\) are assumed \((A)\): NEWLINE\[NEWLINE \qquad f(x,t,z) \in C^1({\overline \Omega} \times [0,T] \times \mathbb{R}), \quad |f(x,t,z)| \leq C_0(\phi(x,t) + |z|^{\alpha}), NEWLINE\]NEWLINE where \(\phi \geq 0\), \(\phi \in L^r(\Omega \times (0,T)))\), \(r > (N + p^-)/p^-\) and \(C_0 >\), \(\alpha \geq 0\) are constants. The main result reads: Let \(u_0 \in L^{\infty}\cap W_0^{1,p(x)}(\Omega)\). Assume that the conditions \((B)\) hold: NEWLINE\[NEWLINE \qquad \alpha < p^- - 1 \quad \text{or }\quad \alpha = p^- - 1, \quad \text{and } \quad |\Omega| \ll 1,NEWLINE\]NEWLINE where \(|\Omega|\) denotes the Lebesgue measure of \(\Omega\). Then there exists a weak solution of the problem under consideration such that NEWLINE\[NEWLINE u \in L^{\infty}(Q_T) \cap L^{\infty}\left(0,T; W^{1,p(x)}_0(\Omega)\right), \quad u_t \in L^2\left(\Omega \times (0,T)\right). NEWLINE\]NEWLINENEWLINENEWLINEIf \(f(x,t,z) \in C^1({\overline \Omega} \times [0,T] \times \mathbb{R})\), then the solution of the problem with \(u \in L^{\infty}(Q_T)\), \(u_t \in L^2\left(\Omega \times (0,T)\right)\) is unique.NEWLINENEWLINEUnder a weaker condition for \(u_0\) and the above-mentioned assumptions \((A)\) and \((B)\) the following assertion holds: There exists a weak solution \(u\) of the problem such that NEWLINE\[NEWLINE u \in L^{\infty}(Q_T)\cap L^{\infty}\left(\epsilon, T; W^{1,p(x)}_0(\Omega)\right), \quad u_t \in L^2\left(\Omega \times (\epsilon,T)\right), NEWLINE\]NEWLINE where \(0 < \epsilon < T\) is a constant.NEWLINENEWLINEIf \(f(x,t,z) \in C^1({\overline \Omega} \times [0,T] \times \mathbb{R})\), then there exists \(T^* > 0\) that the problem has a solution in \(Q_{T^*}\) such that NEWLINE\[NEWLINE u \in L^{\infty}(Q_{T^*})\cap L^{\infty}\left(\epsilon, T^*; W^{1,p(x)}_0(\Omega)\right), \quad u_t \in L^2\left(\Omega \times (\epsilon,T^*)\right). NEWLINE\]