Diffraction problems for quasilinear parabolic systems with boundary intersecting interfaces
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RESEARCH
Open Access
Diffraction problems for quasilinear parabolic systems with boundary intersecting interfaces Qi-Jian Tan* and Chao-Yi Pan *
Correspondence: [email protected] Department of Mathematics, Chengdu Normal University, Chengdu, 611130, P.R. China
Abstract In this paper, we discuss the n-dimensional diffraction problem for weakly coupled quasilinear parabolic system on a bounded domain , where the interfaces k (k = 1, . . . , K – 1) are allowed to intersect with the outer boundary ∂ and the coefficients of the equations are allowed to be discontinuous on the interfaces. The aim is to show the existence of solutions by approximation method. The approximation problem is a diffraction problem with interfaces, which do not intersect with ∂. MSC: 35R05; 35K57; 35K65 Keywords: diffraction problem; quasilinear parabolic system; interface; approximation method
1 Introduction Let be a bounded domain in Rn with boundary ∂ (n ≥ ), and let be partitioned into a finite number of subdomains k (k = , . . . , K ) separated by k , where k , k = , . . . , K – , are interfaces, which do not intersect with each other. For any T > , set QT := × (, T],
ST := ∂ × [, T],
:=
K–
k ,
T := × [, T].
k=
In this paper, we consider the diffraction problem for quasilinear parabolic reactiondiffusion system in the form ⎧ ⎪ ul – Ll (ul ) = g l (x, t, u) ((x, t) ∈ QT ), ⎪ ⎨ t ⎪ ⎪ ⎩
[ul ]T = ,
[alij (x, t, ul )ulxj νi (x)]T = ,
(.)
ul = ψ l (x, t) ((x, t) ∈ ST ∪ { × {}}), l = , . . . , N,
where x = (x , . . . , xn ), u = (u , . . . , uN ), ult := ∂ul /∂t, ulxi := ∂ul /∂xi , ulx := (ulx , . . . , ulxn ), d l Ll ul := aij x, t, ul ulxj + blj x, t, ul ulxj , dxi
l = , . . . , N,
(.)
repeated indices i or j indicate summation from to n, ν(x) := (ν (x), . . . , νn (x)) is the unit normal vector to (the positive direction of ν(x) is fixed in advance), the symbol [·]T © 2013 Tan and Pan; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Tan and Pan Boundary Value Problems 2013, 2013:99 http://www.boundaryvalueproblems.com/content/2013/1/99
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denotes the jump of a quantity across T , and the coefficients alij (x, t, ul ), blj (x, t, ul ) and g l (x, t, u) are allowed to be discontinuous on T . In the following, we refer to the conditions on T in (.) as diffraction conditions. The diffraction problems often appear in different fields of physics, ecology, and technics. In some of them, the interfaces are allowed to intersect with the outer boundary ∂ (see [–]). The linear diffraction problems have been treated by many researchers (see [–]). For the quasilinear parabolic and elliptic diffraction problems, when all of the interfaces k do not intersect with ∂, the existence and uniqueness of the solutions have b
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