On the existence and asymptotic behavior of solutions of nonlinear heat flow with memory (Q1113023)

The existence and asymptotic behavior of solutions to the following history-boundary-value problem are obtained: \[ \frac{\partial}{\partial t}[u(t,x)+\int^{t}_{-\infty}k(t-s)u(s,x)ds]=\sigma (u_ x(t,x))_ x+h(t,x),\quad t\in {\mathbb{R}}^+,\quad x\in (0,1), \] \[ u_ x(t,0)\in \beta_ 0(u(t,0)),\quad -u_ x(t,1)\in \beta_ 1(u(t,1)),\quad t\in {\mathbb{R}}, \] \[ u(t,x)=u_ 0(x),\quad t\in (-\infty,0],\quad x\in (0,1). \] We treat the above problem in \(L^ p\)-spaces \((1<p<\infty)\) by applying the abstract results concerning the nonlinear Volterra equations in Banach spaces. It is noticed that the gradient of \(\sigma\) and the boundedness of \(\beta_ i\) \((i=0,1)\) are deeply concerned each other.