Open Mathematics (May 2023)
Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
Abstract
The Cauchy problem in Rn{{\mathbb{R}}}^{n}, n≥2n\ge 2, for ut=Δu−∇⋅(uS⋅∇v),0=Δv+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∈Rn×nS\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(Rn)∩Lp(Rn){\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 Tmax∈(0,∞]{T}_{\max }\in \left(0,\infty ] and a uniquely determined u∈C0([0,Tmax);BUC(Rn))∩C0([0,Tmax);Lp(Rn))∩C∞(Rn×(0,Tmax))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⋆uv:= \Gamma \star u, and with Γ\Gamma denoting the Newtonian kernel on Rn{{\mathbb{R}}}^{n}, the pair (u,v)\left(u,v) forms a classical solution of (⋆\star ) in Rn×(0,Tmax){{\mathbb{R}}}^{n}\times \left(0,{T}_{\max }), which has the property that ifTmax<∞,then bothlimsupt↗Tmax‖u(⋅,t)‖L∞(Rn)=∞andlimsupt↗Tmax‖∇v(⋅,t)‖L∞(Rn)=∞.\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).
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