Global and blow-up solutions for a class of nonlinear parabolic problems under Robin boundary condition. (English) Zbl 1470.35190

Zhang, Lingling; Wang, Hui

Abstr. Appl. Anal. 2014, Article ID 241650, 9 p. (2014).

Summary: We discuss the global and blow-up solutions of the following nonlinear parabolic problems with a gradient term under Robin boundary conditions: \((b(u))_t = \nabla \cdot(h(t) k(x) a(u) \nabla u) + f(x, u, | \nabla u |^2, t)\), in \(D \times(0, T)\), \((\partial u / \partial n) + \gamma u = 0\), on \(\partial D \times(0, T)\), \(u(x, 0) = u_0(x) > 0\), in \(\overline{D}\), where \(D \subset \mathbb{R}^N (N \geq 2)\) is a bounded domain with smooth boundary \(\partial D\). Under some appropriate assumption on the functions \(f\), \(h\), \(k\), \(b\), and \(a\) and initial value \(u_0\), we obtain the sufficient conditions for the existence of a global solution, an upper estimate of the global solution, the sufficient conditions for the existence of a blow-up solution, an upper bound for “blow-up time,” and an upper estimate of “blow-up rate.” Our approach depends heavily on the maximum principles.

MSC:

35K55	Nonlinear parabolic equations
35B44	Blow-up in context of PDEs
35K20	Initial-boundary value problems for second-order parabolic equations

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