2013
DOI: 10.1007/978-3-642-41515-9_6
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Comparison Principle for Reaction-Diffusion-Advection Problems with Boundary and Internal Layers

Abstract: Abstract. In the present paper we discuss father development of the general scheme of the asymptotic method of differential inequalities and illustrate it applying for some new important cases of initial boundary value problem for the nonlinear singularly perturbed parabolic equations,which are called in applications as reaction-diffusion-advection equations. The theorems which state front motion description and stationary contrast structures formation are proved for parabolic, parabolic-periodic and integro-p… Show more

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Cited by 45 publications
(5 citation statements)
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“…In this paper, we investigate existence and stability of such solution. For the last purpose, we use the asymptotical method of differential inequalities . The last can be generalized to the multidimensional case in contrast to the techniques used in previous articles.…”
Section: Introductionmentioning
confidence: 99%
“…In this paper, we investigate existence and stability of such solution. For the last purpose, we use the asymptotical method of differential inequalities . The last can be generalized to the multidimensional case in contrast to the techniques used in previous articles.…”
Section: Introductionmentioning
confidence: 99%
“…In [27], this method was extended to periodic parabolic boundary value problems. In that work, a fairly effective method was proposed for proving the asymptotic stability of periodic solutions, which was then transferred to the analysis of the asymptotic stability of stationary solutions of initial-boundary value problems for reaction-diffusion-type parabolic equations and then generalized to some more complex classes of reaction-diffusionadvection-type quasilinear equations (see [29,45]). In recent years, this approach has been extended to reaction-diffusion-advection problems with discontinuous nonlinearities and sources (see [46] and references therein).…”
Section: Asymptotic Methods Of Differential Inequalitiesmentioning
confidence: 99%
“…In [25,26], a new method was proposed for proving the existence of solutions of singularly perturbed partial differential equations, called the asymptotic method of differential inequalities. This method turned out to be effective, and both the general scheme of this method was developed (see, e.g., [27][28][29]) and its application to other classes of singularly perturbed problems: initial-boundary value problems for parabolic equations in describing solutions with moving interior layers (fronts), periodic parabolic boundary value problems, boundary and initial-boundary value problems for some classes of integro-differential equations, and some classes of systems.…”
Section: Introductionmentioning
confidence: 99%
“…From the general scheme of the asymptotic method of differential inequalities [13] (for details see also [8,14,16]) we have the result. 2 (x, ε) are defined by the expressions (2.6) and (2.7) as the upper and lower solutions of problem (1.1), are the upper and lower solutions of problem (3.1), ifα(x, 0, ε) ≤ v 0 (x, ε) ≤β(x, 0, ε), that follows from the inequalities α 2 (x, 0, ε) ≤ v 0 (x, ε) ≤ β 2 (x, 0, ε).…”
Section: Local Uniqueness and Stability Of The Solution Of Periodic Pmentioning
confidence: 98%