This paper is concerned with an initial-boundary value problem for a doubly haptotactic oncolytic virotherapy model. Thus far, existing results on the existence and boundedness of solutions to oncolytic virus models with doubly haptotaxis and extracellular matrix remodeling have been largely confined to the two-dimensional setting with linear diffusion. In particular, in three space dimensions, the doubly haptotactic system with linear diffusion requires additional coefficient restrictions and small initial data to guarantee the solvability of classical solutions. The present work aims to remove the constraints of small initial data and narrow parameter regimes in the three-dimensional setting. To this end, we investigate a three-dimensional doubly haptotactic oncolytic virus model in which linear diffusion is replaced by porous medium diffusion. Precisely, we focus on the following model <Equation ID="Equ77"> <EquationSource Format="TEX">$\begin{aligned} \left\{ \begin{array}{lll} u_t=\Delta u^{m}-\nabla \cdot (u\nabla v)+\mu _u u(1-u)-uz,\\ v_t=-(u+w)v+\mu _v v(1-v),\\ w_t=\Delta w^l-\nabla \cdot (w\nabla v)-w+uz,\\ z_t=\Delta z-z-uz+\beta w, \end{array} \right. \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>u</mi> <mi>m</mi> </msup> <mo>-</mo> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mi mathvariant="normal">∇</mi> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msub> <mi>μ</mi> <mi>u</mi> </msub> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi>u</mi> <mi>z</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <msub> <mi>v</mi> <mi>t</mi> </msub> <mo>=</mo> <mo>-</mo> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>+</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mi>v</mi> <mo>+</mo> <msub> <mi>μ</mi> <mi>v</mi> </msub> <mi>v</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <msub> <mi>w</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <msup> <mi>w</mi> <mi>l</mi> </msup> <mo>-</mo> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mrow> <mo stretchy="false">(</mo> <mi>w</mi> <mi mathvariant="normal">∇</mi> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi>w</mi> <mo>+</mo> <mi>u</mi> <mi>z</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <msub> <mi>z</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>z</mi> <mo>-</mo> <mi>z</mi> <mo>-</mo> <mi>u</mi> <mi>z</mi> <mo>+</mo> <mi>β</mi> <mi>w</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>in a smoothly bounded domain. For some slow diffusion case <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$m>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$l>6/5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>l</mi> <mo>></mo> <mn>6</mn> <mo stretchy="false">/</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>, this problem admits a globally defined weak solution for any large initial values. Unlike the approach based on the exponential transformation used in the linear diffusion case, we introduce a new technique that builds on our previous work to handle the solvability issues posed by the porous medium diffusion. The technique relies on delicate foundational energy estimates and matching the estimates of u,  v and w.

Global Solvability for a Three-Dimensional Doubly Haptotactic System with Porous Medium Diffusion

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发表日期：

2026年8月1日

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Chunhua Jin,
Yi Lu,
Xulong Qin,
Langhao Zhou

摘要

This paper is concerned with an initial-boundary value problem for a doubly haptotactic oncolytic virotherapy model. Thus far, existing results on the existence and boundedness of solutions to oncolytic virus models with doubly haptotaxis and extracellular matrix remodeling have been largely confined to the two-dimensional setting with linear diffusion. In particular, in three space dimensions, the doubly haptotactic system with linear diffusion requires additional coefficient restrictions and small initial data to guarantee the solvability of classical solutions. The present work aims to remove the constraints of small initial data and narrow parameter regimes in the three-dimensional setting. To this end, we investigate a three-dimensional doubly haptotactic oncolytic virus model in which linear diffusion is replaced by porous medium diffusion. Precisely, we focus on the following model $\begin{aligned} \left\{ \begin{array}{lll} u_t=\Delta u^{m}-\nabla \cdot (u\nabla v)+\mu _u u(1-u)-uz,\\ v_t=-(u+w)v+\mu _v v(1-v),\\ w_t=\Delta w^l-\nabla \cdot (w\nabla v)-w+uz,\\ z_t=\Delta z-z-uz+\beta w, \end{array} \right. \end{aligned}$ $\begin{matrix} \{\begin{matrix} u_{t} = Δ u^{m} - \nabla \cdot (u \nabla v) + μ_{u} u (1 - u) - u z, \\ v_{t} = - (u + w) v + μ_{v} v (1 - v), \\ w_{t} = Δ w^{l} - \nabla \cdot (w \nabla v) - w + u z, \\ z_{t} = Δ z - z - u z + β w, \end{matrix}) \end{matrix}$ in a smoothly bounded domain. For some slow diffusion case $m>1$ $m > 1$ and $l>6/5$ $l > 6 / 5$ , this problem admits a globally defined weak solution for any large initial values. Unlike the approach based on the exponential transformation used in the linear diffusion case, we introduce a new technique that builds on our previous work to handle the solvability issues posed by the porous medium diffusion. The technique relies on delicate foundational energy estimates and matching the estimates of u, v and w.