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Global solutions for quasilinear parabolic problems. (English) Zbl 1004.35070

The authors are concerned with the Cauchy-Dirichlet problem for the quasilinear parabolic equation \[ u_t - D_i(a^{ij}(X,u)D_j u)=F(X,u,Du) \] in a bounded cylindrical space-time domain \(\Omega\times(0,T)\). Assuming sufficient smoothness of the coefficients \(a^{ij}\) and \(F\) and a uniform lower bound on the minimum eigenvalue of the matrix \([a^{ij}(X,z)]\), the classical theory reduces the question of solvability of the problem to some a priori estimates on any such solution. In particular, if \(F(X,z,p)=O(|p|^2)\) as \(|p|\to \infty\) (uniformly with respect to \(X\) and any bounded range for \(z\)), this theory guarantees that a solution exists once a bound is obtained for the \(L^\infty\) norm of \(u\). The authors are interested in applying an abstract result on the differentiability of the maximum (and minimum) of a function to proving this estimate. Specifically, they show that if \(u\in W^{1,1}([0,T];C(\overline\Omega))\), then the function \(M\) defined by \[ M(t)= \max_{\overline\Omega} u(\cdot,t) \] is differentiable for almost all \(t\) and that (for almost all \(t\)) there is a point \(\zeta(t)\) such that \(M(t)=u(\zeta(t),t)\) and \(M'(t)=u_t(\zeta(t),t)\) with a similar statement for the minimum of \(u\). In fact, this abstract result is the centerpiece of their work. It is also used to prove a priori estimates for some related problems: a weakly coupled system of parabolic equations and an elliptic equation with dynamic boundary condition. Their main new ingredient is that \(F\) can grow fairly rapidly with respect to \(z\), in fact, at an essentially optimal rate.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K55 Nonlinear parabolic equations
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