Global and Blow‐Up Solutions for a Class of Nonlinear Parabolic Problems under Robin Boundary Condition

Lingling Zhang and Hui Wang

Abstract and Applied Analysis, 2014, vol. 2014, issue 1

Abstract: 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 = ∇·(h(t)k(x)a(u)∇u) + f(x, u, |∇u|2, t), in D × (0, T), (∂u/∂n) + γu = 0, on ∂D × (0, T), u(x, 0) = u0(x) > 0, in D¯, where D⊂RN (N≥2) is a bounded domain with smooth boundary ∂D. Under some appropriate assumption on the functions f, h, k, b, and a and initial value u0, 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.

Date: 2014
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https://doi.org/10.1155/2014/241650

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:241650

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