Strong convergence of weighted gradients in parabolic equations and applications to global generalized solvability of cross-diffusive systems

In the first part of the present paper, we show that strong convergence of (v(0 epsilon))(epsilon is an element of(0,1)) in L-1(Omega) and weak convergence of (f(epsilon))(epsilon is an element of(0,1)) in L-loc(1)((Omega) over barx[0,infinity)) not only suffice to conclude that solutions to the initial boundary value problem {v(epsilon t) = Delta v(epsilon) + f(epsilon)(x,t) in Omega x (0, infinity), partial derivative(nu)v(epsilon) = 0 on partial derivative Omega x (0, infinity), v(epsilon)(center dot, 0) = v(0 epsilon) in Omega, which we consider in smooth, bounded domains Omega, converge to the unique weak solution of the limit problem, but that also certain weighted gradients of v epsilon converge strongly in L-loc(2)((Omega) over bar x [0, infinity)) along a subsequence. We then make use of these findings to obtain global generalized solutions to various cross-diffusive systems. {ut = Delta u - chi del center dot (u/v del v) + g(u), u(t) = Delta u - uv, where chi > 0 and g is an element of C-1([0, infinity)) are given, merely provided that (g(0) >= 0 and) -g grows superlinearly. This result holds in all space dimensions and does neither require any symmetry assumptions nor the smallness of certain parameters. Thereby, we expand on a corresponding result for quadratically growing -g proved by Lankeit and Lankeit (Nonlinearity 32(5):1569-1596, 2019).

Automated summary

In the first part of this paper, we establish the convergence properties of certain functions in different function spaces, and use these results to derive solutions to initial boundary value problems and obtain global generalized solutions to cross-diffusive systems. These findings expand on previous research and do not require symmetry assumptions or smallness of certain parameters.