Quasi-linear parabolic partial differential equations with delays in the highest order spatial derivatives.

*(English)*Zbl 0804.35139Parabolic partial differential equations with delays are studied with delays in the highest order derivatives. These are of the form

$${u}_{t}={u}_{xx}+F\left(u(t,x),{u}_{x}(t,x),u(t+\theta ,x),{u}_{x}(t+\theta ,x),{u}_{xx}(t+\theta ,x)\right)$$

where $F$ contains integral and pointwise operators. The approach is based on an abstract evolution equation of the form

$$\dot{u}\left(t\right)+Au\left(t\right)=G\left(t,{u}_{t}(\xb7)\right),\phantom{\rule{2.em}{0ex}}u\left(0\right)=x,\phantom{\rule{2.em}{0ex}}u\left(t\right)=\phi \left(t\right),\phantom{\rule{1.em}{0ex}}\text{a.e.}\phantom{\rule{4.pt}{0ex}}t\in [-r,0)\xb7$$

Firstly it is shown that this equation has a unique mild solution for any $\phi \in {L}_{\text{loc}}^{2}(-r,\infty ;\mathcal{D}\left(A\right))$. Then some regularity and asymptotic properties of the solution are derived. Finally the results are applied to an integral class of delay systems and the linear case is dealt with in some depth.

Reviewer: S.P.Banks (Sheffield)