论文标题
溶液界限到准线性高维趋化性的新结果 - 具有非线性扩散的触觉模型
A new result for boundedness of solutions to a quasilinear higher-dimensional chemotaxis -- haptotaxis model with nonlinear diffusion
论文作者
论文摘要
本文讨论了一个偶联的准趋化性趋化性问题 - 与非线性扩散$$ $$ \ left \ left \ {\ begin {array} {ll} {ll} u_t = \ nabla \ cdot \ cdot（d（d（d（d（u） v） - ξ\ nabla \ cdot（u \ nabla w） +μU（1-u-w），\\ v_t =Δv-v- v +u，\ quad \\ w_t = - $ n $ n $维平滑域中的$$，其中参数$ξ，χ> 0 $，$μ> 0 $。假定扩散性$ d（u）$满足$ d（u）\ geq c_ {d} u^{m-1} $ for All $ u> 0 $，with一些$ c_d> 0 $。在本文中依靠新的能源不等式，证明在$$ m> \ frac {2n} {n+{{{{\ frac {（\ frac {（\ frac {\ max_ {s \ geq1} （χ+ξ\ | w_0 \ | _ {l^\ infty（ω）}）}} {（\ max_ {s \ geq1}λ_0^{\ fr ac {1} {{s} +1}}}（χ+ξ\ | w_0 \ | _ {l^\ infty（ω）}） - μ）_ {+}}}+1） （n+\ frac {\ max_ {s \ geq1}λ_0^{\ frac {1} {1} {{s} +1}}}（χ+| w_0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ | _ {l^\ infty（ω）}）}} {（\ max_ {s \ geq1}λ_0^{\ frac {1} {1} {{s} +1}}} (χ+ξ\|w_0\|_{L^\infty(Ω)})-μ)_{+}}-1)}{N}}}}},$$ and proper regularity hypotheses on the initial data, the corresponding initial-boundary problem possesses at least one global bounded classical solution when $D(0) > 0$ (the case of non-degenerate扩散），而如果，$ d（0）\ geq 0 $（可能是退化扩散的情况），则显示了系统的有限弱解决方案。这扩展了一些作者的最新结果。
This paper deals with a boundary-value problem for a coupled quasilinear chemotaxis--haptotaxis model with nonlinear diffusion $$\left\{\begin{array}{ll} u_t=\nabla\cdot(D(u)\nabla u)-χ\nabla\cdot(u\nabla v)-ξ\nabla\cdot(u\nabla w)+μu(1-u-w),\\ v_t=Δv- v +u,\quad \\ w_t=- vw\\ \end{array}\right. $$ in $N$-dimensional smoothly bounded domains, where the parameters $ξ,χ> 0$, $μ> 0$. The diffusivity $D(u)$ is assumed to satisfy $D(u)\geq C_{D}u^{m-1}$ for all $u > 0$ with some $C_D>0$. Relying on a new energy inequality, in this paper, it is proved that under the conditions $$m>\frac{2N}{N+{{{\frac{(\frac{\max_{s\geq1}λ_0^{\frac{1}{{s}+1}} (χ+ξ\|w_0\|_{L^\infty(Ω)})}{(\max_{s\geq1}λ_0^{\frac{1}{{s}+1}}(χ+ξ\|w_0\|_{L^\infty(Ω)})-μ)_{+}}+1) (N+\frac{\max_{s\geq1}λ_0^{\frac{1}{{s}+1}}(χ+ξ\|w_0\|_{L^\infty(Ω)})}{(\max_{s\geq1}λ_0^{\frac{1}{{s}+1}} (χ+ξ\|w_0\|_{L^\infty(Ω)})-μ)_{+}}-1)}{N}}}}},$$ and proper regularity hypotheses on the initial data, the corresponding initial-boundary problem possesses at least one global bounded classical solution when $D(0) > 0$ (the case of non-degenerate diffusion), while if, $D(0)\geq 0$ (the case of possibly degenerate diffusion), the existence of bounded weak solutions for system is shown. This extends some recent results by several authors.
