Application of the Leray-Schauder degree to investigation of travelling wave solutions of parabolic equations

We study travelling wave solutions of the semilinear parabolic system of equations https://www.w3.org/1998/Math/MathML"> ∂ u ∂ t = α Δ u + r ( x ′ ) ∂ u ∂ x 1 + F ( u ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math915.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> (1) in the infinite cylinder https://www.w3.org/1998/Math/MathML"> Ω ⊂ R m , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/inline-math866.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> with the boundary condition https://www.w3.org/1998/Math/MathML"> ∂ u ∂ n | ∂ Ω = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math916.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> (2) Here https://www.w3.org/1998/Math/MathML"> Ω = D × R 1 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/inline-math867.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> , D is a bounded domain in R^{m -1} with a smooth boundary, x ₁ is a coordinate along the axis of the cylinder, x' = (x ₂,x_m ), w = (w ₁, ...,w_n ), F = (F ₁,...,F _n), a is a constant symmmetric positive-definite matrix, r(x') is a scalar function. Travelling wave solution of this system is a solution of the form

https://www.w3.org/1998/Math/MathML"> u ( x , t ) = w ( x 1 - c t , x 2 , ... , x n ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/inline-math868.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

Here c is a constant, the wave velocity. The function w(x) is a solution of the following problem https://www.w3.org/1998/Math/MathML"> a Δ w + ( r ( x ′ ) + c ) ∂ w ∂ x 1 + F ( w ) = 0 , ∂ w ∂ n | ∂ Ω = 0.. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math917.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> (3) We assume also that it has the limits at the infinity: https://www.w3.org/1998/Math/MathML"> lim x 1 → ± ∞ w ( x ) = w ± , w + ≠ w - . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003062271/3a9a5b62-3a88-4cc5-923a-97624426ebfd/content/math918.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> (4) Existence of multidimensional waves was studied in [1], [6], [8] (see the complete bibliography in [17], [23]).