Abstract
This paper is concerned with traveling curved fronts of bistable reaction–diffusion equations with nonlinear convection in a two-dimensional space. By constructing super- and subsolutions, we establish the existence of traveling curved fronts. Furthermore, we show that the traveling curved front is globally asymptotically stable.
Highlights
In this paper, we consider traveling wave solutions of the following reaction–diffusion equations with a nonlinear convection term: ut + g(u) y = u + f (u), (x, y) ∈ R2, t > 0, (1.1)where f is the nonlinear reaction term and (g(u))y is the nonlinear convection term
From Theorem 1.1, we find that traveling curved front v satisfying (1.10) and (1.12) is unique
This paper is organized as follows: In Sect. 2, we prove the existence of the traveling curved front v by constructing an appropriate supersolution of (1.10)
Summary
We consider traveling wave solutions of the following reaction–diffusion equations with a nonlinear convection term: ut + g(u) y = u + f (u), (x, y) ∈ R2, t > 0,. For the sake of convenience, in the sequel we always denote (Uθ (·), cθ ) and sθ by (U(·), c) and s, respectively It follows from Ninomiya and Taniguchi [43] that there exists a unique function φ(x) with asymptotic lines y = m∗|x| satisfying s φxx 1 + φx. Theorem 2.3 There exists a traveling wave solution u(x, y, t) = v∗(x, y + st) of (1.1) satisfying (1.10) and lim sup v∗(x, z) – v–(x, z).
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