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

Abstract

AbstractIn the first part of the present paper, we show that strong convergence of $$(v_{0 \varepsilon })_{\varepsilon \in (0, 1)}$$ ( v 0 ε ) ε ∈ ( 0 , 1 ) in $$L^1(\Omega )$$ L 1 ( Ω ) and weak convergence of $$(f_{\varepsilon })_{\varepsilon \in (0, 1)}$$ ( f ε ) ε ∈ ( 0 , 1 ) in $$L_{\text {loc}}^1({{\overline{\Omega }}} \times [0, \infty ))$$ L loc 1 ( Ω ¯ × [ 0 , ∞ ) ) not only suffice to conclude that solutions to the initial boundary value problem $$\begin{aligned} {\left\{ \begin{array}{ll} v_{\varepsilon t} = \Delta v_\varepsilon + f_\varepsilon (x, t) &{} \text {in }\Omega \times (0, \infty ), \\ \partial _\nu v_\varepsilon = 0 &{} \text {on }\partial \Omega \times (0, \infty ), \\ v_\varepsilon (\cdot , 0) = v_{0 \varepsilon } &{} \text {in }\Omega , \end{array}\right. } \end{aligned}$$ v ε t = Δ v ε + f ε ( x , t ) in Ω × ( 0 , ∞ ) , ∂ ν v ε = 0 on ∂ Ω × ( 0 , ∞ ) , v ε ( · , 0 ) = v 0 ε in Ω , which we consider in smooth, bounded domains $$\Omega $$ Ω , converge to the unique weak solution of the limit problem, but that also certain weighted gradients of $$v_\varepsilon $$ v ε converge strongly in $$L_{\text {loc}}^2({{\overline{\Omega }}} \times [0, \infty ))$$ L loc 2 ( Ω ¯ × [ 0 , ∞ ) ) along a subsequence. We then make use of these findings to obtain global generalized solutions to various cross-diffusive systems. Inter alia, we establish global generalized solvability of the system $$\begin{aligned} {\left\{ \begin{array}{ll} u_t = \Delta u - \chi \nabla \cdot (\frac{u}{v} \nabla v) + g(u), \\ v_t = \Delta v - uv, \end{array}\right. } \end{aligned}$$ u t = Δ u - χ ∇ · ( u v ∇ v ) + g ( u ) , v t = Δ v - u v , where $$\chi > 0$$ χ > 0 and $$g \in C^1([0, \infty ))$$ g ∈ C 1 ( [ 0 , ∞ ) ) are given, merely provided that ($$g(0) \ge 0$$ g ( 0 ) ≥ 0 and) $$-g$$ - 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 $$-g$$ - g proved by Lankeit and Lankeit (Nonlinearity 32(5):1569–1596, 2019).