Yuta Wakasugi

A note on the lifespan of solutions to the semilinear damped wave equation

This paper concerns estimates of the lifespan of solutions to the semilinear damped wave equation. We give upper estimates of the lifespan for the semilinear damped wave equation with variable coefficients in all space dimensions.

Critical Exponent for the Semilinear Wave Equation with Scale Invariant Damping

In this paper we consider the critical exponent problem for the semilinear damped wave equation with time-dependent coefficients. We treat the scale invariant cases. In this case the asymptotic behavior of the solution is very delicate and the size of coefficient plays an essential role. We shall prove that if the power of the nonlinearity is greater than the Fujita exponent, then there exists a unique global solution with small data, provided that the size of the coefficient is sufficiently large. We shall also prove some blow-up results even in the case that the coefficient is sufficiently small.

Small data global existence for the semilinear wave equation with space–time dependent damping

Abstract In this paper we consider the critical exponent problem for the semilinear wave equation with space–time dependent damping. When the damping is effective, it is expected that the critical exponent agrees with that of only the space dependent coefficient case. We shall prove that there exists a unique global solution for small data if the power of nonlinearity is larger than the expected exponent. Moreover, we do not assume that the data are compactly supported. However, it is still open whether there exists a blow-up solution if the power of nonlinearity is smaller than the expected exponent.

Remarks on an elliptic problem arising in weighted energy estimates for wave equations with space-dependent damping term in an exterior domain

This paper is concerned with weighted energy estimates and di usion phenomena for the initial-boundary problem of the wave equation with space-dependent damping term in an exterior domain. In this analysis, an elliptic problem was introduced by Todorova and Yordanov. This attempt was quite useful when the coeffcient of the damping term is radially symmetric. In this paper, by modifying their elliptic problem, we establish weighted energy estimates and di usion phenomena even when the coeffcient of the damping term is not radially symmetric.