Existence of solutions to chemotaxis dynamics with Lipschitz diffusion and superlinear growth

Yokota, Tomomi; Yoshino, Noriaki

doi:10.1016/j.jmaa.2014.04.069

Cited by 8 publications

(16 citation statements)

References 13 publications

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“…We approximate D ( r ) by

D_{ϵ} (r) : = D (r + ϵ), D_{ϵ, λ} (r) : = \{\begin{matrix} D_{ϵ} (r) & (r \leq λ), \\ D_{ϵ} (λ) + D_{ϵ}^{'} (λ) (r - λ) & (r \geq λ), \end{matrix}

where 0 < ϵ < 1 and 1 < λ < ∞ . Then D ϵ , λ satisfies

0 < D^{'} (ϵ) \leq D_{ϵ, λ}^{'} (r) \leq D^{'} (λ + 1) < \infty,

and thus we can apply the result for or , and confirm existence of approximate solutions. However, needs the smallness assumption for

{(∥∥, b_{0})}_{L^{2} (normalΩ)}

, and it depends on the constant D 0 in , which depends on the parameter ϵ in this case.…”

Section: Introduction and Main Resultsmentioning

confidence: 53%

“…One of the difficulties in the proof of existence is to deal with the chemotaxis term in the space of unbounded and nonsmooth functions such as L 2 (Ω). To overcome the difficulty we will apply our previous result obtained in via the theory for nonlinear m ‐accretive operators.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Therefore we cannot argue the limit transition because existence of solutions is not guaranteed for small ϵ . On the other hand, removed the smallness assumption for

{(∥∥, b_{0})}_{L^{2} (normalΩ)}

, which ensures existence of approximate solutions for any ϵ and allows to the limit transition argument. The proof of Theorem is based on estimates for solutions of .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…/. To overcome the difficulty we will apply our previous result obtained in [2] via the theory for nonlinear maccretive operators.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Then there exists T ϵ , λ >0 such that has a unique solution ( b ϵ , λ , c ϵ , λ ) such that 0≤ b ϵ , λ ∈ C ([0, T ϵ , λ ]; H ) ∩ L 2 (0, T ϵ , λ ; V ) ∩ H 1 (0, T ϵ , λ ; V ′),0≤ c ϵ , λ ∈ C ([0, T ϵ , λ ]; D ( A Δ )).Proof Let 0 < ϵ < 1 and 1 < λ < ∞ . Since

0 < D^{'} (ϵ) \leq D_{ϵ, λ}^{'} (r) \leq D^{'} (λ) < \infty (r > 0)

and ( r 1 , r 2 )↦ K ϵ , λ ( r 1 , r 2 ) r 1 is Lipschitz continuous on

{double-struckR}^{2}

in view of Lemma , it follows from , Theorem 1.1] that there exists a unique solution ( b ϵ , λ , c ϵ , λ ) of . Note that the initial data is nonnegative, and so are b ϵ , λ , c ϵ , λ .…”

Section: Approximate Systemmentioning

confidence: 98%

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Local and global existence of solutions to a quasilinear degenerate chemotaxis system with unbounded initial data

Yokota

Yoshino

2015

Math Methods in App Sciences

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“…We approximate D ( r ) by

D_{ϵ} (r) : = D (r + ϵ), D_{ϵ, λ} (r) : = \{\begin{matrix} D_{ϵ} (r) & (r \leq λ), \\ D_{ϵ} (λ) + D_{ϵ}^{'} (λ) (r - λ) & (r \geq λ), \end{matrix}

where 0 < ϵ < 1 and 1 < λ < ∞ . Then D ϵ , λ satisfies

0 < D^{'} (ϵ) \leq D_{ϵ, λ}^{'} (r) \leq D^{'} (λ + 1) < \infty,

and thus we can apply the result for or , and confirm existence of approximate solutions. However, needs the smallness assumption for

{(∥∥, b_{0})}_{L^{2} (normalΩ)}

, and it depends on the constant D 0 in , which depends on the parameter ϵ in this case.…”

Section: Introduction and Main Resultsmentioning

confidence: 53%

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Therefore we cannot argue the limit transition because existence of solutions is not guaranteed for small ϵ . On the other hand, removed the smallness assumption for

{(∥∥, b_{0})}_{L^{2} (normalΩ)}

, which ensures existence of approximate solutions for any ϵ and allows to the limit transition argument. The proof of Theorem is based on estimates for solutions of .…”