On stable space dependent stationary solutions of a competition system with diffusion
A quasilinear parabolic system of two equations is considered in connection with the following initial boundary value problem: \[ \partial_t u = a_1\Delta u + u(b_1 - c_1u - d_1v),\quad \partial_t v = v(b_2 - c_2v - d_2u) \quad\text{in } (0,+\infty)\times \Omega, \] \[ u=0\quad\text{in } (0,+\infty)\times \partial \Omega, \quad u=u_0,\ v=v_0\quad\text{in }\{0\} \times \Omega. \] Here \(\Omega \subset \mathbb R^n\) is an open bounded set with smooth boundary \(\partial \Omega\), \(a_i,b_i,c_i,d_i\) \((i=1,2)\) are positive constants and \(u_0\), \(v_0\) are given nonnegative functions. The authors investigate existence, uniqueness and asymptotic stability of the following elliptic problem: \[ a_1\Delta u + u(b_1 - c_1u - d_1v) = 0,\quad v(b_2 - c_2v - d_2u) = 0\quad\text{in } \Omega, \ u=0\quad\text{in } \partial \Omega. \] The main purpose of the paper is to prove what happens when arbitrarily small diffusion is introduced. For this purpose the elliptic problem \[ a_1\Delta u + u(b_1 - c_1u- d_1v)=0,\ \varepsilon_1\Delta u+v(b_2-c_2v-d_2u)=0\quad\text{in }\Omega, \ u=v=0\quad\text{on }\partial \Omega, \] where \(0<\varepsilon \ll a_1\) is considered. The bifurcation of nonnegative stationary solutions is studied together with their stability character and their dependence on the ``space'' variables.
scientific article; zbMATH DE number 3846078