On a quasilinear parabolic integrodifferential equation (Q1343074)

The author considers the nonlinear Volterra integrodifferential equation \(u_ t - a* \text{div} h(\text{grad} u) = a*g\), where \(x \in \mathbb{R}^ n\), \(t \geq 0\) and where the initial function \(u(0,x) = w(x)\) is given. The kernel \(a\) satisfies \(a \in L^ 1_{\text{loc}} (\mathbb{R}^ +)\) and the parabolic condition \(\text{Re}\widetilde a (\omega) \geq q | \text{Im} \widetilde a (\omega) |\) some \(q > 0\) and \(\omega \in \mathbb{R}\). Suppose that for some \(p > 4\), \[ g \in Y = L^ 2 (\mathbb{R}^ + (\mathbb{R}^ +; L^ 2(\mathbb{R}^ n)) \cap L^{p, \infty} (\mathbb{R}^ +; H^{n-1} (\mathbb{R}^ n)), \] where \(L^{p, \infty} (\mathbb{R}^ +; X) {\buildrel {\text{def}} \over =} \left \{f : \sup_{T \geq 0} \int^{T+1}_ T \| f \|^ p_ X < \infty \right\}\). It is shown that for \(\| g \|_ Y + \| \text{grad} w \|_{H^ n (\mathbb{R}^ n)}\) sufficiently small there exists a solution \(u\), defined on \(\mathbb{R}^ + \times \mathbb{R}^ n\) and satisfying \(u_ t\), \(\Delta u \in Y\).