Conservation laws in mathematical biology (Q432654)

There are posed the mathematical models of the different biological processes containing conservation laws of the form NEWLINE\[NEWLINE \partial {\mathbf u}/\partial t +\text{div} (V\,{\mathbf u}) = {\mathbf F}(t,{\mathbf x},{\mathbf u}),\leqno(1) NEWLINE\]NEWLINE where \({\mathbf x} = (x_1,\dots,x_n)\), \({\mathbf u} = (u_1(t,{\mathbf x}),\dots,u_k(t,{\mathbf x}))\), \({\mathbf F} = (F_1(t,{\mathbf x},{\mathbf u}),\dots,F_k(t,{\mathbf x},{\mathbf u}))\), \(V\) is a matrix with the elements \(V_{i,j}(t,{\mathbf x},{\mathbf u})\), \(F_i(t,{\mathbf x},{\mathbf u})\), \(V_{ij}(t,{\mathbf x},{\mathbf u})\) are nonlinear and/or nonlocal functions of \({\mathbf u}\).NEWLINENEWLINEThese models consider such things, for instance, as bacterial transmission, tumor growth, drug treatment, moving of the sells, cell differentiation, wound healing.NEWLINENEWLINEThe considered models are the mathematical problems for the equations (1) coupled to the other systems, such as ordinary differential equations, elliptic equations, parabolic equations with suitable initial and boundary conditions in the fixed or sometimes in the unknown (with a free boundary) domains.NEWLINENEWLINEThere are given the author's results of the existence, uniqueness of the solutions of these problems, asymptotic behavior of the solutions as \(t\to \infty\) or as a small parameter \(\epsilon\) goes to zero, behavior of the free boundary.