This paper is concerned with the nonlinear damped wave equation on a measure space with a self-adjoint operator, instead of the standard Laplace operator. Under a certain decay estimate on the corresponding heat semigroup, we establish the linear est
imates which generalize the so-called Matsumura estimates, and prove the small data global existence of solutions to the damped wave equation based on the linear estimates. Our approach is based on a direct spectral analysis analogous to the Fourier analysis. The self-adjoint operators treated in this paper include some important examples such as the Laplace operators on Euclidean spaces, the Dirichlet Laplacian on an arbitrary open set, the Robin Laplacian on an exterior domain, the Schrodinger operator, the elliptic operator, the Laplacian on Sierpinski gasket, and the fractional Laplacian.
The Cauchy problem for the Hardy-Henon parabolic equation is studied in the critical and subcritical regime in weighted Lebesgue spaces on the Euclidean space $mathbb{R}^d$. Well-posedness for singular initial data and existence of non-radial forward
self-similar solution of the problem are previously shown only for the Hardy and Fujita cases ($gammale 0$) in earlier works. The weighted spaces enable us to treat the potential $|x|^{gamma}$ as an increase or decrease of the weight, thereby we can prove well-posedness to the problem for all $gamma$ with $-min{2,d}<gamma$ including the Henon case ($gamma>0$). As a byproduct of the well-posedness, the self-similar solutions to the problem are also constructed for all $gamma$ without restrictions. A non-existence result of local solution for supercritical data is also shown. Therefore our critical exponent $s_c$ turns out to be optimal in regards to the solvability.
The purpose in this paper is to determine the global behavior of solutions to the initial-boundary value problems for energy-subcritical and critical semilinear heat equations by initial data with lower energy than the mountain pass level in energy s
paces associated with self-adjoint operators satisfying Gaussian upper bounds. Our self-adjoint operators include the Dirichlet Laplacian on an open set, Robin Laplacian on an exterior domain, and Schrodinger operators, etc.
This paper is devoted to the study of gradient estimates for the Dirichlet problem of the heat equation in the exterior domain of a compact set. Our results describe the time decay rates of the derivatives of solutions to the Dirichlet problem. Appli
cations of these estimates to bilinear type commutator estimates for Laplace operator with Dirichlet boundary condition in exterior domain are discussed too.
The purpose of this paper is to give a definition and prove the fundamental properties of Besov spaces generated by the Neumann Laplacian. As a by-product of these results, the fractional Leibniz rule in these Besov spaces is obtained.
The purpose of this paper is to establish bilinear estimates in Besov spaces generated by the Dirichlet Laplacian on a domain of Euclidian spaces. These estimates are proved by using the gradient estimates for heat semigroup together with the Bony pa
raproduct formula and the boundedness of spectral multipliers.
This paper is devoted to giving definitions of Besov spaces on an arbitrary open set of $mathbb R^n$ via the spectral theorem for the Schrodinger operator with the Dirichlet boundary condition. The crucial point is to introduce some test function spa
ces on $Omega$. The fundamental properties of Besov spaces are also shown, such as embedding relations and duality, etc. Furthermore, the isomorphism relations are established among the Besov spaces in which regularity of functions is measured by the Dirichlet Laplacian and the Schrodinger operators.
