On a quasilinear parabolic integrodifferential equation

Abstract

We consider the nonlinear Volterra integrodifferential equation $$ u_t - a* \text{div}\, h(\,\text{grad}\, u)=a*g, $$ where $x\in \mathbb{R}$, $t\geq 0$ and where the initial function $u(0,x)=w(x)$ is given. The kernel $a$ satisfies $a\in L_{\text{loc}} ^1(\mathbb{R}^{+})$ and the parabolicity condition $\mathfrak{R} \tilde a(\omega )\geq q | \Im \tilde a (\omega) | $ for some $q > 0$ and $\omega \in \mathbb{R}$. We suppose that for some $p> 4$ $$ g\in Y= L^2(\mathbb{R};L^2(\mathbb{R}^n)) \cap L^{p,\infty}(\mathbb{R}; H^{n-1}(\mathbb{R}^n)), $$ where $L^{p,\infty}(\mathbb{R};X)\overset{\rm {def}} \to = \{f : \sup_{T\geq 0}\int ^{T+1}_T \|f\|^p_X \! < \! \infty \}.$ It is shown that for $\|g\|_Y+\|\text{grad\,} w \|_{H^n(\mathbb{R}^n)}$ sufficiently small there exists a solution $u$, defined on $\mathbb{R} \times \mathbb{R}$ and satisfying $u_t,\Delta u\in Y$.