Existence and Optimization of the Critical Speed for Travelling Front Solutions with Convection in Unbounded Cylinders

For n > 1, we consider a reaction-diffusion equationut = Δu + α(y)∇ · G(u) + f(u), (0.2)in an unbounded cylinder Ω := R×D, where D ⊂ Rn−1 is a smooth bounded domain, with a presence of a convection term, under both Neumann and Dirichlet boundary conditions on ∂Ω. For both types of boundary condition, we consider two different forms of convection term, namely : α(y)∇·G(u) and∇ · (α(y)G(u)). The reaction term f is “monostable”. In both Neumann and Dirichlet cases, we prove that there exists a critical speed c⋆ ∈ R such that there exists a travelling front solution of the form u(x, t) = w(x1 −ct, y) with speed c if and only if c ≥ c⋆, where x1 is the coordinate corresponding to theaxis of the cylinder. The critical speed c⋆ often plays an important role for monostable problems by characterizing the long-time behaviour of the initial value problem. The existence of travelling waves for all c ≥ c⋆ is typical of monostable problems such as the prototype Fisher-KPP equation.We give a min-max formula for the speed c⋆. For both types of boundary conditions, we prove that c⋆ is bounded below by a quantity c′ which is related to a certain eigenvalue problem, associated with the linearized problem around 0. Note that under Dirichlet boundary conditions, an extra assumption is needed to ensure that c′ exists, namely, f′(0) has to be greater than the principal eigenvalue of the linearized operator. We discuss two special cases where the equality c⋆ = c′ holds. Under both Neumann and Dirichlet boundary conditions, the first special case is when G = (G1, 0, · · ·, 0), assuming the so-called KPP condition for f and that α(y)G′ 1(u) ≥ α(y)G′ 1(0), for all y ∈ D and all u ∈ (0, 1). The second case is treated only under Neumann boundary conditions : when G′ 1(0) = 0, assuming the KPP condition for f, and that α(y)G′ 1(u) ≥ 0, for all y ∈ D and u ∈ (0, 1). Note that in that case, we give an explicit formula : c⋆ = c′ = 2 p f′(0). Under Dirichlet boundary conditions, we highlight the influence of the domain D, the reaction term f and the convection term α(y)∇ · G(u) on the critical speed c⋆. In the special case where G = (G1, 0, ···, 0), using that c⋆ = c′, we use the eigenvalue problem related to c′ to establish some optimization results for c⋆.

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