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${L}^{p}$-${L}^{q}$ estimates of solutions to the damped wave equation in 3-dimensional space and their application. (English) Zbl 1023.35078
Summary: It has been asserted that the damped wave equation has the diffusive structure as $t\to \infty$. 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 ${L}^{p}$-${L}^{q}$ 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 ${L}^{p}$-${L}^{q}$ 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.

##### MSC:
 35L70 Nonlinear second-order hyperbolic equations 35L15 Second order hyperbolic equations, initial value problems 35B33 Critical exponents (PDE) 35B45 A priori estimates for solutions of PDE
##### Keywords:
power nonlinearity; Fujita exponent; time global existence