Abstract
The $L^∞$-energy method is developed so as to handle nonlinear parabolic systems with convection and hysteresis effect. The system under consideration originates from a biological model where the hysteresis and convective effects are taken into account in the evolution of species. Some results for the existence of local and global solutions as well as the uniqueness of solution are presented.
Highlights
The present paper deals with the following system of nonlinear PDE’s with convection and hysteresis effect:∂σ − ∇ · (∇σ + λ(σ)) + ∂IU (σ) ∂t g(σ, U ) (x, t) ∈ QT = Ω × (0, T ), (1) ∂ui ∂t − ∇ · (∇ui + μi(ui)) =
The system (1)-(2) could be considered as a biological model where the hysteresis and convective effects are taken into account in the evolution of the species
The hysteresis effect is described by the subdifferential term ∂IU (·) of the indicator function
Summary
The present paper deals with the following system of nonlinear PDE’s with convection and hysteresis effect:. The system (1)-(2) could be considered as a biological model where the hysteresis and convective effects are taken into account in the evolution of the species. Equations (1) and (2) correspond to the evolution of 1 + m biological species, for example, prey and m - types of predators, where σ and U = (u1, · · · , um) are the population densities of the prey and the predators respectively. The hysteresis effect is described by the subdifferential term ∂IU (·) of the indicator function. L∞-energy method, parabolic system, convection, hysteresis, subdifferential operator.
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