\(L^ \infty\)-estimates for nonlinear parabolic equations with natural growth conditions
Let \(\Omega\) be a bounded domain in \(\mathbb{R}^ N\) with boundary \(\partial \Omega\) and for \(0<T< \infty\) let \(\Omega_ T\) denote the cylindrical domain \(\Omega\times (0,T]\). Let also \(\Gamma\) be the parabolic boundary of \(\Omega_ T\). Assume that the boundary \(\partial \Omega\) satisfies the property of positive geometric density, i.e. there exist \(c>0\) and \(r_ 0\) such that for each \(x_ 0\in \partial \Omega\), for every ball \(B_ r (x_ 0)\) centered in \(x_ 0\) with radius \(r\leq r_ 0\), the measure of the intersection between \(\Omega\) and \(B_ r (x_ 0)\) is greater than \(cr^ N\). Consider the boundary value problem for \(u\in C(0,T; L^ 2 (\Omega))\cap L^ p (0,T; W^{1,p} (\Omega))\): \[ u_ t- \text{Div } a(x,t,u, Du)= b(x,t,u, Du) \quad \text{in }\Omega_ T, \qquad u|_ \Gamma= f\in L^ \infty (\Gamma), \tag{1} \] where \(p\) is a number greater than 1 and the p.d.e. satisfies the structure conditions \[ a(x,t,u, Du)\cdot Du\geq C_ 0 | Du|^ p- \varphi(x,t), \qquad | b(x,t,u, Du)|\leq C_ 1| Du|^ p+ \varphi. \] Here \(C_ i\), \(i=0,1\), are given positive constants and the nonnegative function \(\varphi\) satisfies \(\varphi\in L^ q (\Omega_ T)\) where \(1/q= (1- \kappa_ 0) p\kappa_ 0/ (N+p)\in (0,1]\). Our main result is the following Theorem. Let \(u\) be a qualitatively bounded weak solution of (1) in \(\Omega_ T\). There exists a constant \(C\) that can be determined quantitatively a priori only in terms of the data, such that \(\| u\|_{\infty, \Omega_ T}\leq \max\{2 \| f\|_{\infty, \Gamma}; C\}\).
scientific article; zbMATH DE number 510092
scientific article