Abstract
We study the blow-up and global solutions for a class of quasilinear parabolic problems with Robin boundary conditions. By constructing auxiliary functions and using maximum principles, the sufficient conditions for the existence of blow-up solution, an upper bound for the “blow-up time,” an upper estimate of the “blow-up rate,” the sufficient conditions for the existence of global solution, and an upper estimate of the global solution are specified.
Highlights
In this paper, we are going to investigate the blow-up and global solutions of the following quasilinear parabolic problem with Robin boundary conditions:(g (u))t = ∇ ⋅ (a (u, t) b (x) ∇u) + h (t) f (u) in D × (0, T), ∂u ∂n + γu = on ∂D × (0, T), (1)u (x, 0) = u0 (x) > 0 in D, where D ⊂ RN (N ≥ 2) is a bounded domain with smooth boundary ∂D, ∂/∂n represents the outward normal derivative on ∂D, γ is a positive constant, u0 is the initial value, T is the maximal existence time of u, and D is the closure of D
Passing to the limit as t → +∞ in (31) yields which contradicts assumption (9). This shows that u must blow up in a finite time t = T
Α = 1, the following two assumptions (i)a and (ii)a can guarantee that inequality (23) holds
Summary
We are going to investigate the blow-up and global solutions of the following quasilinear parabolic problem with Robin boundary conditions:. Many authors have studied the blow-up and global solutions of nonlinear parabolic problems (see, for instance, [3– 14]). Some authors discussed blow-up phenomena for parabolic problems with Robin boundary conditions and obtained a lot of interesting results (see [18–24] and the references cited therein). Blow-up and global solutions for parabolic equations reflect the unsteady state and steady state of heat conduction process, respectively. By constructing completely different auxiliary functions with those in [15–17] and technically using maximum principles, we obtain some existence theorems of blow-up solution, an upper bound of “blow-up time,” an upper estimates of “blowup rate,” the existence theorems of global solution, and an upper estimate of the global solution. A few examples are presented in Section 4 to illustrate the applications of the abstract results
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