Abstract
In this paper, we establish the Fujita type theorem for a homogeneous Neumann outer problem of the coupled quasilinear convection–diffusion equations and formulate the critical Fujita exponent. Besides, the influence of diffusion term, reaction term, and convection term on the global existence and the blow-up property of the problem is revealed. Finally, we discuss the large time behavior of the solution to the outer problem in the critical case and describe the asymptotic behavior of the solution.
Highlights
In this paper, we consider the critical Fujita exponent of the following coupled quasilinear convection–diffusion equations: ∂u = ∂t um + κ x |x|2 · ∇ um|x|λvp, x ∈ Rn\B1, t > 0, (1) ∂v = ∂t vm
In [9], the authors studied the Fujita type theorem for the outer problem of the following coupled nonlinear diffusion equations with convective terms:
We prove that the critical Fujita exponent is
Summary
We consider the critical Fujita exponent of the following coupled quasilinear convection–diffusion equations:. In [9], the authors studied the Fujita type theorem for the outer problem of the following coupled nonlinear diffusion equations with convective terms:. For the global existence of the problem solution, we use the method of constructing the self-similar solution and the comparison principle to prove our conclusion. The following existence theorem and the comparison principle to problem (1)–(4) play an important role in proving our main results. To prove the existence of a nontrivial global solution to problem (1)–(4), we introduce the following form of self-similar supersolutions to system (1) and (2): u(x, t) = (t + 1)–αU (t + 1)–β |x| , x ∈ Rn\B1, t ≥ 0,.
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