Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type

Abstract

Abstract The Cauchy problem in R n {{\mathbb{R}}}^{n} , n ≥ 2 n\ge 2 , for u t = Δ u − ∇ ⋅ ( u S ⋅ ∇ v ) , 0 = Δ v + u , ( ⋆ ) \begin{array}{r}\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{u}_{t}=\Delta u-\nabla \cdot \left(uS\cdot \nabla v),\\ 0=\Delta v+u,\end{array}\right.\hspace{2.0em}\hspace{2.0em}\hspace{2.0em}\left(\star )\end{array} is considered for general matrices S ∈ R n × n S\in {{\mathbb{R}}}^{n\times n} . A theory of local-in-time classical existence and extensibility is developed in a framework that differs from those considered in large parts of the literature by involving bounded classical solutions. Specifically, it is shown that for all non-negative initial data belonging to BUC ( R n ) ∩ L p ( R n ) {\rm{BUC}}\left({{\mathbb{R}}}^{n})\cap {L}^{p}\left({{\mathbb{R}}}^{n}) with some p ∈ [ 1 , n ) p\in \left[1,n) , there exist T max ∈ ( 0 , ∞ ] {T}_{\max }\in \left(0,\infty ] and a uniquely determined u ∈ C 0 ( [ 0 , T max ) ; BUC ( R n ) ) ∩ C 0 ( [ 0 , T max ) ; L p ( R n ) ) ∩ C ∞ ( R n × ( 0 , T max ) ) u\in {C}^{0}\left(\left[0,{T}_{\max });\hspace{0.33em}{\rm{BUC}}\left({{\mathbb{R}}}^{n}))\cap {C}^{0}\left(\left[0,{T}_{\max });\hspace{0.33em}{L}^{p}\left({{\mathbb{R}}}^{n}))\cap {C}^{\infty }\left({{\mathbb{R}}}^{n}\times \left(0,{T}_{\max })) such that with v ≔ Γ ⋆ u v:= \Gamma \star u , and with Γ \Gamma denoting the Newtonian kernel on R n {{\mathbb{R}}}^{n} , the pair ( u , v ) \left(u,v) forms a classical solution of ( ⋆ \star ) in R n × ( 0 , T max ) {{\mathbb{R}}}^{n}\times \left(0,{T}_{\max }) , which has the property that if T max < ∞ , then both limsup t ↗ T max ‖ u ( ⋅ , t ) ‖ L ∞ ( R n ) = ∞ and limsup t ↗ T max ‖ ∇ v ( ⋅ , t ) ‖ L ∞ ( R n ) = ∞ . \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}{T}_{\max }\lt \infty ,\hspace{1.0em}\hspace{0.1em}\text{then both}\hspace{0.1em}\hspace{0.33em}\mathop{\mathrm{limsup}}\limits_{t\nearrow {T}_{\max }}\Vert u\left(\cdot ,t){\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{n})}=\infty \hspace{1.0em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1.0em}\mathop{\mathrm{limsup}}\limits_{t\nearrow {T}_{\max }}\Vert \nabla v\left(\cdot ,t){\Vert }_{{L}^{\infty }\left({{\mathbb{R}}}^{n})}=\infty . An exemplary application of this provides a result on global classical solvability in cases when ∣ S + 1 ∣ | S+{\bf{1}}| is sufficiently small, where 1 = diag ( 1 , … , 1 ) {\bf{1}}={\rm{diag}}\hspace{0.33em}\left(1,\ldots ,1) .