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Travelling wavefronts of reaction-diffusion equations in cylindrical domains. (English) Zbl 0816.35058
From the introduction: This paper is concerned with the existence and uniqueness (up to translation) of some bounded solutions of \begin{aligned} \Delta &u+ cu_ x+ f(u)=0 \quad \text{in } \mathbb{R}\times \Omega,\tag{1}\\ &u=0 \quad \text{at } \mathbb{R}\times \partial \Omega,\tag{2}\\ &u (x,\cdot) \to v_ \pm \quad \text{pointwise as }x\to \pm\infty, \tag{3}\end{aligned} that may be viewed (after a suitable change of variables) as travelling wave solutions of the parabolic problem $$\partial u/\partial t= \Delta u+ f(u)$$ in $$\mathbb{R}\times \Omega$$, with the boundary condition (2) and an appropriate initial condition. Here, $$\Omega\subset \mathbb{R}^{n-1}$$ $$(n\geq 2)$$ is a bounded domain, with a $$C^{2,\alpha}$$ boundary, for some $$\alpha>0$$ if $$n\leq 2$$, and the spatial coordinates are written as $$(x,y)$$, where $$x=x_ 1$$ and $$y= (x_ 2,\dots, x_ n)$$. The constant $$c$$ is the wave velocity that is to be determined, $$f: \mathbb{R}\to \mathbb{R}$$ is a $$C^ 2$$-function, and $$v_ -$$ and $$v_ +$$ are solutions of $$\Delta v+ f(v)= 0$$ in $$\Omega$$, $$v=0$$ at $$\partial \Omega$$. We assume that $$v_ - (y)< v_ + (y)$$ for all $$y\in \Omega$$, and we consider only the solution of (1)–(3) such that $$v_ - (y)\leq u(x,y)\leq v_ + (y)$$ for all $$(x,y)\in \mathbb{R}\times \Omega$$.

##### MSC:
 35K57 Reaction-diffusion equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35J65 Nonlinear boundary value problems for linear elliptic equations
##### Keywords:
wavefronts; travelling wave solutions
Full Text:
##### References:
 [1] Agmon S., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I 12 pp 623– (1959) · Zbl 0093.10401 [2] Adams, R.A. 1975. ”Sobolev Spacesrdquo;”. Academic Press. [3] Protter, M. and Weinberger, H. 1977. ”Maximum Principles in Differential Equations”. Prentice Hall. · Zbl 0153.13602 [4] Kirchgässner K., Wave-solutions of reversible systems and applications 45 pp 113– (1982) [5] Bona J.L., Finite-amplitude steady waves in stratified fluids 62 pp 389– (1983) · Zbl 0491.35049 [6] Amick C.J., Nonlinear elliptic eigenvalue problems on an infinite strip. Global theory of bifurcation and asymptotic bifurcation 262 pp 313– (1983) · Zbl 0489.35067 [7] Esteban M.J., Non-linear elliptic problems in strip-like domains: Symmetry of positive vortex rings 7 pp 365– (1983) · Zbl 0513.35035 [8] Amick C.J., Semilinear elliptic eigenvalue problems on an infinite strip, with an application to stratified fluids 11 pp 441– (1984) · Zbl 0568.35076 [9] Craig W., Symmetry of solitary waves 13 pp 603– (1988) [10] Berestycki, H. and Nirenberg, L. 1990.Some qualitative properties of solutions of semi-linear elliptic equations in cylindrical domains, 115–164. Academic Press. in Analysis, etc (volume dedicated to J. Moser), P.H. Rabinowitz and E. Zehnder eds · Zbl 0705.35004 [11] Gardner R., Existence of multidimensional travelling wave solutions of an initial-boundary value problem 61 pp 335– (1986) · Zbl 0549.35066 [12] Berestycki H., A semi-linear elliptic equation in a strip arising in a two-dimensional flame propagation model 396 pp 14– (1989) · Zbl 0658.35036 [13] Berestycki H., Multidimensional travelling-wave solutions of a flame propagation model 111 pp 33– (1990) · Zbl 0711.35066 [14] Berestycki H., Travelling front solutions of semilinear equations in n-dimensions (1992) · Zbl 0780.35054 [15] Vega J.M., Multidimensional travelling wavefronts in a model from Combustion theory and in related problems (1992) [16] Vega J.M., On the uniqueness of multidimensional travelling fronts of some semilinear equations (1992) [17] Berestycki H., Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations 5 pp 237– (1988) · Zbl 0698.35031 [18] Agmon S., Properties of solutions of ordinary differential equations in Banach space 16 pp 121– (1963) · Zbl 0117.10001 [19] Pazy A., Asymptotic expansions of solutions of ordinary differential equations in Hilbert space 24 pp 193– (1967) · Zbl 0147.12303 [20] Vega J.M., The asymptotic behavior of the solutions of some semilinear elliptic equations in cylindrical domains (1992) [21] Buonincontri S., Multidimensional travelling wave solutions to reaction-diffusion equations 43 pp 261– (1989) · Zbl 0713.65076 [22] Hagstrom T., Asymptotic analysis of disipative waves with applications to their numerical simulation (1990) [23] Parra I.E., Multiple solutions of some semilinear elliptic equations in slender cylindrical domains (1992) · Zbl 0816.35034 [24] Smoller, J. 1983. ”ldquo;Shock Waves and Reaction-Diffusion Equationsrdquo;”. Springer-Verlag. [25] Hale, J.K. 1988. ”Asymptotic Behavior of Dissipative Systems”. American Mathematical Society · Zbl 0642.58013 [26] Henry, D. 1981. ”Geometric Theory of Semilinear Parabolic Equations”. Vol. 840, Springer-Verlag. Lecture Notes in Math. · Zbl 0456.35001 [27] Kolmogorov A.N., A study of the equation of diffusion with increase in the quantity of matter 1 pp 1– (1937) [28] Cohen D.S., Nonlinear boundary value problems suggested by chemical reactor theory 7 pp 217– (1970) · Zbl 0201.43102 [29] Smoller J., Global bifurcation of steady solutions 39 pp 269– (1981) · Zbl 0425.34028 [30] Aris, R. 1975. ”The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts”. Vol. 1, Clarendon Press. · Zbl 0315.76051
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