Boundedness of solutions to a virus infection model with saturated chemotaxis

TL;DR

This paper proves the global existence and boundedness of solutions for a virus infection model with chemotaxis, under specific conditions on the parameters and domain dimensions, ensuring the model's mathematical stability.

Contribution

It establishes the boundedness of classical solutions to a complex virus infection model with chemotaxis in arbitrary dimensions, extending previous results to more general conditions.

Findings

Solutions exist globally and are bounded under certain parameter conditions.

Boundedness holds for any sufficiently regular initial data.

Results apply to domains of arbitrary dimension.

Abstract

We show global existence and boundedness of classical solutions to a virus infection model with chemotaxis in bounded smooth domains of arbitrary dimension and for any sufficiently regular nonnegative initial data and homogeneous Neumann boundary conditions. More precisely, the system considered is \[ \begin{cases}\begin{split} & u_t=\Delta u - \nabla\cdot(\frac{u}{(1+u)^{\alpha}}\nabla v) - uw + \kappa - u, \\ & v_t=\Delta v + uw - v, \\ & w_t=\Delta w - w + v, \end{split}\end{cases} \] with $\kappa\ge 0$, and solvability and boundedness of the solution are shown under the condition that \[ \begin{cases} \alpha > \frac 12 + \frac{n^2}{6n+4}, &\text{if } \quad 1 \leq n \leq 4 \\ \alpha > \frac {n}4, &\text{if } \quad n \geq 5. \end{cases} \]