Pointwise estimates via parabolic potentials for a class of doubly nonlinear parabolic equations with measure data
In the space-time cylinder \(E_T:=E\times (0,T)\), where \(E\subset\mathbb R^n\) is an open bounded set, \(n\geq 2\), \(T>0\), the author considers the following Coauchy-Dirichlet problem \[ \begin{cases} u_t-\operatorname{div}\big(\mathbf A(x,t,u,Du)\big)=\mu, \quad \text{in}\quad E_T,\\ u=0,\quad \text{on}\quad \Gamma_T, \end{cases}\tag{1} \] where \(\mu\) is a non-negative Radon measure on \(E_T\) with finite total mass \(\mu(E_T)<\infty\) and \(\Gamma_T:=\big(\overline E\times \{0\}\big)\bigcup\big(\partial E\times(0,T)\big)\). The nonlinear parabolic equation \[ u_t-\operatorname{div}\big(| u|^{m-1}\,| Du|^{p-2}\,Du\big)=\mu, \quad \text{in}\quad E_T, \] is the model example for the equation from (1). The operator \(\mathbf A:E_T\times\mathbb R\times\mathbb R^n\mapsto\mathbb R\) satisfy the following ellipticity conditions: \textbf{(A1)} \(\mathbf A\) is measurable with respect to \((x,t)\in E_T\) for all \((u,\xi)\in \mathbb R\times\mathbb R^n\) and continuous with respect to \((u,\xi)\in \mathbb R\times\mathbb R^n\) for almost every \((x,t)\in E_T\);\,\, \textbf{(A2)}\, \(\big(\mathbf A(x,t,u,\xi),\xi\big)\geq C_0\,| u|^{m-1}\,|\xi|^{p}\);\,\, \textbf{(A3)}\, \(|\mathbf A(x,t,u,\xi)|\leq C_1 |u|^{m-1}\,|\xi|^{p-1}\), where \(C_0>0\), \(C_1>0\), \(p\geq 2\) and \(m>1\) in both conditions \textbf{(A2)} and \textbf{(A3)}. The main result of the paper is the following. Suppose that \(\mathbf A\) satisfies conditions \textbf{(A1)}--\textbf{(A3)} and \(\mu\in L^{q_1,q_2}_{\text{loc}}\big(E; L^{q_1,\infty}_{\text{loc}}(0,T)\big)\), for \(q_1=\frac{n+p}{p}\) and \(q_2=\frac{n+p}{n(p-1)+p}\). Then any weak solution \(u\) to the problem (1) is locally uniformly bounded in \(E_T\). The author also defines the nonlinear parabolic potentials for the equation (1) and in terms of these potentials establishes the pointwise estimates for weak solutions to the problem (1).
scientific article; zbMATH DE number 6945635
scientific article