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Existence and non-existence of global solutions for a higher-order semilinear parabolic system. (English) Zbl 1100.35039
The paper is devoted to the Cauchy problem for a weakly coupled higher-order semilinear parabolic system. Higher-order semilinear and quasilinear heat equations appear in a number of areas such as thin film theory, flame propagation, bi-stable phase transition, and higher-order diffusion. It is shown that under certain conditions on the system’s structure, solutions with small initial data exist globally in time. The authors also adduce conditions, under which every solution with initial data having positive average value is not global. With the help of the semigroup form of the solution the authors also deduce an \(L^\infty\)-decay estimate.

35K30 Initial value problems for higher-order parabolic equations
35K45 Initial value problems for second-order parabolic systems
35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
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