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Global existence and nonexistence theorems for quasilinear evolution equations of formally parabolic type. (English) Zbl 0891.35062

The authors study global existence and nonexistence of solutions of abstract quasilinear evolution problems involving potential operators. Using energy methods, they show nonexistence of global solutions provided the energy of the initial function is negative (or \(\ll0\)). The global existence result is based on suitable structure conditions on the corresponding potentials. The abstract theory is illustrated by examples of the type \(|u_t|^{m-2}u_t-a\nabla\cdot(|\nabla u|^{q-2}\nabla u)=|u|^{p-2}u\).

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K65 Degenerate parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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