Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type

Abstract

The article deals with nonlinear second-order evolutionary partial differential equations (PDEs) of the parabolic type with a reasonably general form. We consider the case of PDE degeneration when the unknown function vanishes. Similar equations in various forms arise in continuum mechanics to describe some diffusion and filtration processes as well as to model heat propagation in the case when the properties of the process depend significantly on the unknown function (concentration, temperature, etc.). One of the exciting and meaningful classes of solutions to these equations is diffusion (heat) waves, which describe the propagation of perturbations over a stationary (zero) background with a finite velocity. It is known that such effects are atypical for parabolic equations; they arise as a consequence of the degeneration mentioned above. We prove the existence theorem of piecewise analytical solutions of the considered type and construct exact solutions (ansatz). Their search reduces to the integration of Cauchy problems for second-order ODEs with a singularity in the term multiplying the highest derivative. In some special cases, the construction is brought to explicit formulas that allow us to study the properties of solutions. The case of the generalized porous medium equation turns out to be especially interesting as the constructed solution has the form of a soliton moving at a constant velocity.

Multidimensional Diffusion-Wave-Type Solutions to the Second-Order Evolutionary Equation

“…A new example is constructed, which is an analog of the classic example by S.V. Kovalevskaya [41] and generalizes the earlier result [36].…”

“…Previously, we have considered problem (5), (12) for ν = 0, α = 0, and u 0 = ρ n , n ∈ N and figured out that series (13) ends if n = 1, converges for n = 2, and diverges if n ≥ 3 [36].…”

“…One can find a detailed review of studies of the porous medium equation and its generalizations up to 2007 in the fundamental monograph [2]. Also, our previous articles [36,37] give detailed bibliographic reviews.…”

“…We have repeatedly examined zero-front solutions in the context of constructing exact solutions [36,38,39], proving the existence and uniqueness theorems for initial boundary problems that initiate a heat wave [36,38,39], and developing numerical algorithms [37,40]. Moreover, the article [38] provides the immediate background for this study in terms of problem formulation, while the research methods draw from [36,39] to address a higherdimensional problem.…”

“…We also prove a theorem for the existence of analytical solutions to the boundary problem for Equation (1). Unlike [36,38,39], we perform the hodograph transformation in order to overcome analytical difficulties. A new example is constructed, which is an analog of the classic example by S.V.…”