关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

Preface 1. Fourier series and Fourier transforms 2. Distributions 3. Elliptic equations (fundamental theory) 4. Initial value problems (Cauchy problems) 5. Evolution equations 6. Hyperbolic equations 7. Semi-linear hyperbolic equations 8. Green's functions and spectra Supplementary remarks Guide to the literature Bibliography Symbols Index.

The theory of partial differential equations

.We study abstract evolution equations with nonlinear damping terms and source terms, including as a particular case a nonlinear wave equation of the type $ \ba{cl} u_{tt}-\Delta u+ b|u_t|^{m-2}u_t=c|u|^{p-2}u, & (t,x)\in [0,T)\times\Omega,\\[6pt] u(t,x)=0, & (t,x)\in [0,T)\times\partial \Omega,\\[6pt] u(0,\cdot)=u_0\in H_0^1(\Omega), \quad u_t(0,\cdot)=v_0\in L^2(\Omega),\es& \ea $ where $ 0$, $p>2$, $m>1$. We prove a global nonexistence theorem for positive initial value of the energy when $ 1 1$.

Global nonexistence theorems for a class of evolution equations with dissipation

It has been asserted that the damped wave equation has the diffusive structure as t→∞. In this paper we consider the Cauchy problem in 3-dimensional space for the linear damped wave equation and the corresponding parabolic equation, and obtain the Lp−Lq estimates of the difference of each solution, which represent the assertion precisely. Explicit formulas of the solutions are analyzed for the proof. The second aim is to apply the Lp−Lq estimates to the semilinear damped wave equation with power nonlinearity. If the power is larger than the Fujita exponent, then the time global existence of small weak solution is proved and its optimal decay order is obtained.

Lp-Lq estimates of solutions to the damped wave equation in 3-dimensional space and their application

We consider the N-dimensional Cauchy problem in R N for a semilinear damped wave equation with a power-type nonlinearity | u | p  For a noncompactly supported initial data, which has a small energy, we shall derive a global in time existence result in the case when the power of the nonlinear term satisfies 1 + 2 / N p ⩽ N / [ N - 2 ] +  This generalizes a previous result due to Todorova–Yordanov (J Differential Equations 174 (2001) 464–489), which dealt with a solution in the framework of compactly supported initial data

Global existence of solutions for semilinear damped wave equations in RN with noncompactly supported initial data

Wave equations with strong damping in Hilbert spaces

Optimal energy decay estimates will be derived for weak solutions to the Cauchy problem in R N ðN ¼ 1;2;3Þ of dissipative wave equations, which have lower power nonlinearities juj p� 1 u satisfying 1 þ 2=N < paN=½N � 2� þ .

Ryo Ikehata

Papers

Global existence of solutions for semilinear damped wave equations in RN with noncompactly supported initial data

Wave equations with strong damping in Hilbert spaces

Decay estimates of solutions for dissipative wave equations in R^N with lower power nonlinearities

Some remarks on the wave equations with nonlinear damping and source terms

Asymptotic profiles for wave equations with strong damping