Hiroki Tanabe

An L^{p}-approach to singular linear parabolic equations in bounded domains

Singular means here that the parabolic equation is not in normal form neither can it be reduced to such a form. For this class of problems, following the operator approach used in [1], we prove global in time existence and uniqueness theorems related to (spatial) -spaces. Various improvements to [2], [3] are given.

Singular evolution integro-differential equations with kernels defined on bounded intervals

We study linear singular first-order integro-differential Cauchy problems in Banach spaces. The adjective “singular” means here that the integro-differential equation is not in normal form neither can it be reduced to such a form. We generalize some existence and uniqueness theorems proved in [5] for kernels defined on the entire half-line R + to the case of kernels defined on bounded intervals removing the strict assumption that the kernel should be Laplace-transformable. Particular attention is paid to single out the optimal regularity properties of solutions as well as to point out several explicit applications relative to singular partial integro-differential equations of parabolic and hyperbolic type.

SOLUTIONS OF QUASILINEAR WAVE EQUATION WITH STRONG AND NONLINEAR VISCOSITY

We study a class of quasilinear wave equations with strong and nonlinear viscosity. By using the perturbation method for semilinear parabolic equations, we have established the fundamental results on exis- tence, uniqueness and continuous dependence on data of weak solutions.

Degenerate Integrodifferential Equations of Volterra Type in Banach Space

This paper is concerned with the following degenerate integrodifferential equations of parabolic type.

Functional analytic methods for partial differential equations

\left\{ {\begin{array}{*{20}{c}} {\frac{d}{{dt}}\left( {M\left( t \right)u\left( t \right)} \right) + L\left( t \right)u\left( t \right) + \int_{0}^{t} {K\left( {t,s} \right)u\left( s \right)ds = f\left( t \right),0 < t \leqslant T,} } \\ {M(t)u(t){|_{{t = 0}}} = M(0){u_{0}}.} \\ \end{array} } \right.

On the equations of evolution in a Banach space

(1)