Global Weak Solutions in a Haptotactic Cross-Diffusion System Modeling Oncolytic Virotherapy with Nonlinear Diffusion.

This paper discusses an initial-boundary value problem for a doubly haptotactic cross-diffusion system arising from the oncolytic virotherapy u t = Δ u m - ∇ · (u ∇ v) + μ u (1 - u) - u z , x ∈ Ω , t > 0 , v t = - (u + w) v , x ∈ Ω , t > 0 , w t = Δ w - ∇ · (w ∇ v) - w + u z , x ∈ Ω , t > 0 , z t = D Δ z - z - u z + β w , x ∈ Ω , t > 0 , ∂ u m ∂ ν - u ∂ v ∂ ν = ∂ w ∂ ν - w ∂ v ∂ ν = ∂ z ∂ ν = 0 , x ∈ ∂ Ω , t > 0 , u (x , 0) = u 0 , v (x , 0) = v 0 , w (x , 0) = w 0 , z (x , 0) = z 0 , x ∈ Ω , <graphic mime-subtype="GIF" href="245_2025_10232_Article_Equ100.gif"></graphic> in a smooth bounded domain Ω ⊂ R N (N = 1 , 2) with m > 1 , β > 0 , μ > 0 , and D > 0 . We prove that for any large initial datum, the problem admits a global ‘very’ weak solution for any m > 1 . [ABSTRACT FROM AUTHOR]