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

The Cauchy Problem

Wave equations with time-dependent dissipation II. Effective dissipation

We study the Cauchy problem for the semilinear structural damped wave equation with source term 








with σ ∈ (0,1] in space dimension n ≥ 2 and with a positive constant μ. We are interested in the influence of σ on the critical exponent pcrit in | f(u) | ≈ | u | p. This critical exponent is the threshold between global existence in time of small data solutions and blow-up behavior for some suitable range of p. Our results are optimal for σ = 1 ∕ 2. Copyright © 2013 John Wiley & Sons, Ltd.

Semilinear structural damped waves

A shift in the Strauss exponent for semilinear wave equations with a not effective damping

Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation, II

The authors study the Cauchy problem for the semi-linear damped wave equation 

$$u_{tt} - \Delta u + b\left( t \right)u_t = f\left( u \right), u\left( {0,x} \right) = u_0 \left( x \right), u_t \left( {0,x} \right) = u_1 \left( x \right)$$

in any space dimension n ≥ 1. It is assumed that the time-dependent damping term b(t) > 0 is effective, and in particular tb(t) → ∞ as t → ∞. The global existence of small energy data solutions for |f(u)| ≈ |u|p in the supercritical case of $p > \tfrac{2} {n}$ and $p \leqslant \tfrac{n} {{n - 2}}$ for n ≥ 3 is proved.

https://www.camath.fudan.edu.cn/camb/ch/reader/create_pdf.aspx?file_no=34B303&flag=1

Semi-linear wave equations with effective damping

We study the global existence of small data solutions for Cauchy problem for the semi-linear structural damped wave equation with source term.

Michael Reissig

Papers

Semilinear structural damped waves

A shift in the Strauss exponent for semilinear wave equations with a not effective damping

Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation, II

Semi-linear wave equations with effective damping

Semi-linear structural damped waves