The global existence for diffusion equations with small delay

In this article, we prove that for scalar dissipative delay-diffusion equations u_t - Δu = f(u(t),u(t - τ)) with a small delay, there exists a global mild solution. Our main theory is the fundamental theorem on sectorial operators in. We choose A = -Δ,W = L²(Ω),D(Δ^½) = H₀¹(Ω). Imposing some condition on the nonlinear f, we first make use of the fixed point principle to prove the local existence and uniqueness theorem for mild solutions of the Initial-Bounded Value Problem. Then taking the inner-product in L² (Ω) of both sides of equation with - Δu, we utilize The Energy Method to prove the global existence.