The existence of proper weak solutions of the Cauchy problem constituted by the Navier-Stokes-Fourier system which characterizes the incompressible homogeneous Newtonian fluids under thermal effects is studied. We call proper weak solutions such weak solutions that verify some local energy inequalities in analogy with the suitable weak solutions for the Navier-Stokes equations.

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In this paper, we consider the local well-posedness of the wave equation with null form nonlinearities under the periodic boundary condition. First, we prove the local existence based on some L^p-based Sobolev like Space $\hat{H}$^s,p, where s > 1/2 + d/p′ for d ≥ 2, s ≥ 1/2, for d = 1. Second, we prove the local existence for randomized initial data in H^s with s > 1/2 and d ≥ 2.

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This paper shows Hölder type maximal regularity for the Cauchy problem of linear abstract parabolic evolution equations in Banach spaces. In author's monograph [20, Chapter 3], such regularity has already been established in the case when the domains of the linear operators appearing in the equations are temperately varying with respect to the time variable. This paper is devoted to handling the more general case when the domains are completely varying. Maximal regularity is known to be an essential property of linear parabolic differential equations.

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In this paper, we investigated the generalized Sturm-Liouville boundary value problem in Banach spaces. Using the upper or lower solution method, we obtained some sufficient conditions of the existence results without making any compactness type assumption. The iterative methods and formula of estimating the error were also established. Finally, we gave an example to demonstrate the application of our results.

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We consider an inverse problem involving a parabolic equation which contains an integral operator of convolution type. We show that we can approximate its kernel by using three readings of a certain weighted average of the solution. In case the kernel is known to be band limited then the three measurements are enough for its full recovery.

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We prove the existence of global decaying solutions to the Cauchy problem for the wave equation of Klein-Gordon type with a nonlinear dissipation and a derivative nonlinearity. To derive required estimates of solutions we employ a delicate 'loan' method.

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The present paper is concerned with linear integral equations of the form x(t) = ∫^t_−∞ K(t−s)x(s)ds, where K(s) is a matrix measurable in s, and satisfies some boundedness condition. We propose a dynamical-systems approach to studying the behavior of the equations by treating them as functional equations with infinite delay. As a result we obtain a decomposition of the phase space corresponding to a set of normal eigenvalues of the generator of solution semigroups associated with the equations. Moreover, we establish a representation formula (which is called "a variation-of-constants formula" in the phase space) for solutions of nonhomogeneous integral equations, together with a decomposed formula based on the decomposition of the phase space. Finally, we apply the formula to investigate the admissibility of function spaces with respect to these equations.

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