Monotone traveling waves for reaction-diffusion equations involving the curvature operator

Abstract

We study the existence of monotone traveling waves u(t, x) = u(x + ct), connecting two equilibria, for the reaction-diffusion PDE u t = (for the reaction term f (u) (among which we have the so-called types A, B, and C), we show that, concerning the admissible speeds, the situation presents both similarities and differences with respect to the classical case. We use a first order model obtained after a suitable change of variables. The model contains a singularity and therefore has some features which are not present in the case of linear diffusion. The technique used involves essentially shooting arguments and lower and upper solutions. Some numerical simulations are provided in order to better understand the features of the model.

Heteroclinic traveling fronts for reaction-convection-diffusion equations with a saturating diffusive term

“…We close this Introduction with a plan of the paper. In Section 2 we present the general procedure to reduce equation (1.1) to the first order; we here adapt to the presence of the nonlinear term D(u) the technique already exploited, for instance, in the papers [22,37]. Essentially, it consists in a suitable change of variables heavily relying on the fact that we search for monotone solutions.…”

Monotone wave fronts for $(p, q)$-Laplacian driven reaction-diffusion equations

“…Since v is strictly monotone, the map z → v(z) is invertible, so that we can take v as the new independent variable and write z = z(v). Setting φ(v) = v (z(v)) and proceeding similarly, e.g., as in [7,10], the differential equation in (2.2) is rewritten as…”

“…The first question we are interested in regards the possible speeds of propagation of the traveling waves, namely the values c ∈ R for which (1.2) has a monotone solution, which are called admissible speeds. In the classical case p = q = 2, this problem has been extensively studied (see, e.g., the references in [4,10]) and it has been shown that the set of the admissible speeds is an unbounded interval, whose lower endpoint is called critical speed and is denoted by c * . The value of c * depends on the behavior of the reaction term f : if f is always below its tangent line in 0, it is c * = 2 f (0).…”

Stability properties and dynamics of solutions to viscous conservation laws with mean curvature operator

“…[13,15,26]), where they were essentially motivated by the need to restore the finiteness of the energy along sharp interfaces, thus allowing discontinuous solutions. Indeed, some differences with the linear diffusion case are already present at the level of traveling fronts, since discontinuous steady states here appear naturally (see, e.g., [11,13,15]). As for the convective term, we assume without loss of generality that f ∈ C 2 (R) is such that f (0) = 0; the main example we have in mind is a Burgers-type convection (namely, f (u) = u 2 /2), even if many of the results we state can be extended to more general choices.…”