The regularity of weak solutions for certain <i>n</i>-dimensional strongly coupled parabolic systems

Abstract This paper is concerned with the n -dimensional strongly coupled parabolic systems triangular form in cylinder Ω × ( 0 , T stretchy="false">] \Omega \times (0,T] . We investigate L 2 {L}^{2} and Hölder regularity of derivatives weak solutions ( u 1 ) \left({u}_{1},{u}_{2}) for following two cases: one that boundedness {u}_{1} {u}_{2} has not been shown existence result solutions; other or solutions. By using difference ratios Steklov averages methods various estimates, we prove if a solution system, then any accent="false">′ ⊂ width="0.33em" ^{\prime} \subset \hspace{-0.3em}\subset \hspace{0.33em}\Omega t ∈ t^{\prime} \in \left(0,T) , {u}_{1},{u}_{2} belong to C α width="0.1em" / ¯ [ ] {C}^{\alpha ,\alpha \text{/}2}\left(\bar{\Omega }^{\prime} \left[t^{\prime} ,T]) W {W}_{2}^{2,1}\left(\Omega (t^{\prime} under certain conditions, + {C}^{2+\alpha ,1+\alpha stronger assumptions. Applications these results are given ecological models cross-diffusion.