Galerkin Characteristics Method for Convection-diffusion Problems with Memory Terms

(1.1) ∂tb(x, u) + div(F̄ (t, x, u)− k∇u) = f(t, x, u, s), s(t, x) = ∫ t 0 K(t, z)ψ(u(z, x))dz in Ω × (0, T ], T < ∞, Ω ⊂ R is a bounded domain, ∂Ω ∈ C, see [26]. If Ω is convex, then ∂Ω is assumed to be Lipschitz continuous. We consider a Dirichlet boundary condition (1.2) u(t, x) = 0 on I × ∂Ω, I = (0, T ], together with the initial condition (1.3) u(0, x) = u0(x) x ∈ Ω. We assume 0 < ε ≤ ∂sb(x, s) ≤ M < ∞, k > 0 and suppose that f is sublinear in u, s and ψ(z) is sublinear in z. The convection term F̄ is Lipschitz continuous in u. The mathematical model (1.1)-(1.3) is motivated by contaminant transport in porous media intensively studied in the last years, see [4, 9, 10, 11, 19, 20, 21, 1]