An asymptotic supnorm estimate for solutions of 1-D systems of convection–diffusion equations

We show how to use the LpLp–LqLq approach to obtain fundamental estimates for the spatial supnorm values of solutions u(x,t)=(u1(x,t),…,um(x,t))u(x,t)=(u1(x,t),…,um(x,t)) to general systems of convection–diffusion equations of the form∂ui∂t+∂∂xfi(x,t,u1,…,um)+∂∂xgi(t,ui)=μi(t)∂2ui∂x2,1≤i≤m, with initial data u(⋅,0)∈Lp0(R)∩L∞(R)u(⋅,0)∈Lp0(R)∩L∞(R) for some 1≤p0<∞1≤p0<∞, where μi(t)>0μi(t)>0, given arbitrary f=(f1,…,fm)f=(f1,…,fm), g=(g1,…,gm)g=(g1,…,gm) such that |f(x,t,u)|≤B(t)|u||f(x,t,u)|≤B(t)|u| for all x∈Rx∈R, t≥0t≥0, u∈Rmu∈Rm, where B∈C0([0,∞[)B∈C0([0,∞[).