Transition Fronts of Combustion Reaction Diffusion Equations in $$\mathbb {R}^{N}$$RN

Transition semi-wave solutions of reaction diffusion equations with free boundaries

“…Suppose that p c * (x 0 ) = 0, i.e., q c * (x 0 ) = 0. This and the first equation of (7) yield that f (q c * (x 0 )) = 0. Hence q ≡ q c * (x 0 ) is also a solution of q − c * q + f (q) = 0, which contradicts the uniqueness of trajectory of (27).…”

“…On the other hand, a solution of (7) gives a semi-wave (q(ct − x), ct) of (2) with f = f (u). Note that the equation (7) may have no solution in general. Our first result of this paper is as following: (6) and that (c * , q c * ) is a solution of (7).…”

“…Note that the equation (7) may have no solution in general. Our first result of this paper is as following: (6) and that (c * , q c * ) is a solution of (7). Let (u(t, x), h(t)) be a bounded transition semi-wave of (2) which connects 1 and 0.…”

“…Suppose that (c, q c ) is another solution of (7). We may, without loss of generality, assume that c < c * .…”

“…In [5,6], Berestycki and Hamel introduced a generalized concept of the traveling wave, named the transition wave, which is still a special kind of the entire solutions and describes a general class of wave-like solutions for reaction diffusion equations in general heterogeneous media. Then there were many further works on transition waves for random dispersal, see [4,7,8,13,24,25,33,36,44] and references therein. Existence of transition waves of the heterogeneous Fisher-KPP equations with nonlocal dispersal under some assumptions can be found in [31] and [37].…”