TRAVELLING WAVE PHENOMENA IN HETEROGENEOUS TISSUES

Abstract

Disturbances (or information) propagating in heterogeneous biological tissues (or other media) are often modeled by a partial differential equation of the form $$ u''(t,x) +D(x)u'(t,x) +A(x)u(t,x)=f(t,x), $$ for $ 0<t<T<\infty, x\in\Omega\subset R^n$, where $A$ denotes an elliptic partial differential operator and $D$ represents the ``damping term, where energy dissipates from the system. The solutions are in general damped travelling waves. In order to model physical situations where the spatio-temporal parameters change with time, it is natural to consider a more general model where the coefficients in the operators are allowed to change in time. Hence equations of the form $$ u''(t,x) +D(t,x)u'(t,x) +A(t,x)u(t,x)=f(t,x), $$ arise naturally. However, this ``strong formulation is well-known to cause serious numerical problems, so in this paper we study the abstract hyperbolic model with time dependent ``damping rate given by $$ \langle u''(t),v\rangle_{V',V}+d(t;u'(t),v)+a(t;u(t),v)=\langle f(t),v\rangle_{V',V} \,, $$ where $V\subset V_D\subset H\subset V'_D\subset V'$ are Hilbert spaces with continuous and dense injections, here $H$ is identified with its dual and $\langle\cdot,\cdot\rangle$ denote the various duality products. We show that this very general model allows a unique solution under natural conditions on the time-dependent sesquilinear forms $a(t;\cdot,\cdot):V\times V\to C$ and $d(t;\cdot,\cdot):V_D\times V_D\to C$ and that the solution depends continuously on the data of the problem. This ensures good convergence properties of approximating Galerkin schemes for the numerical solution of the problem.