arXiv subjects

Yuta Wakasugi

Publications and source records attributed to Yuta Wakasugi.

47 records · Page 3Linked to original sources

arXiv2016

Estimates of lifespan and blow-up rates for the wave equation with a time-dependent damping and a power-type nonlinearity

We study blow-up behavior of solutions for the Cauchy problem of the semilinear wave equation with time-dependent damping. When the damping is effective, and the nonlinearity is subcritical, we show the blow-up rates and the sharp lifespan estimates of solutions. Upper estimates are proved by an ODE argument, and lower estimates are given by a method of scaling variables.

math.AP↗

arXiv2016

Scaling variables and asymptotic profiles for the semilinear damped wave equation with variable coefficients

We study the asymptotic behavior of solutions for the semilinear damped wave equation with variable coefficients. We prove that if the damping is effective, and the nonlinearity and other lower order terms can be regarded as perturbations, then the solution is approximated by the scaled Gaussian of the corresponding linear parabolic problem. The proof is based on the scaling variables and energy estimates.

math.AP↗

arXiv2016

The Cauchy problem for the nonlinear damped wave equation with slowly decaying data

We study the Cauchy problem for the nonlinear damped wave equation and establish the large data local well-posedness and small data global well-posedness with slowly decaying initial data. We also prove that the asymptotic profile of the global solution is given by a solution of the corresponding parabolic problem, which shows that the solution of the damped wave equation has the diffusion phenomena. Moreover, we show blow-up of solution and give the estimate of the lifespan for a subcritical nonlinearity. In particular, we determine the critical exponent for any space dimension.

math.AP↗

arXiv2016

Movement of time-delayed hot spots in Euclidean space

We investigate the shape of the solution of the Cauchy problem for the damped wave equation. In particular, we study the existence, location and number of spatial maximizers of the solution. Studying the shape of the solution of the damped wave equation, we prepare a decomposed form of the solution into the heat part and the wave part. Moreover, as its another application, we give $L^p$-$L^q$ estimates of the solution.

math.AP↗

arXiv2016

Diffusion phenomena for the wave equation with space-dependent damping in an exterior domain

In this paper, we consider the asymptotic behavior of solutions to the wave equation with space-dependent damping in an exterior domain. We prove that when the damping is effective, the solution is approximated by that of the corresponding heat equation as time tends to infinity. Our proof is based on semigroup estimates for the corresponding heat equation and weighted energy estimates for the damped wave equation. The optimality of the decay late for solutions is also established.

math.AP↗

arXiv2013

On diffusion phenomena for the linear wave equation with space-dependent damping

In this paper, we prove the diffusion phenomenon for the linear wave equation with space-dependent damping. We prove that the asymptotic profile of the solution is given by a solution of the corresponding heat equation in the $L^2$-sense.

math.AP↗

arXiv2013

Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables

In this paper, we give a small data blow-up result for the one-dimensional semilinear wave equation with damping depending on time and space variables. We show that if the damping term can be regarded as perturbation, that is, non-effective damping in a certain sense, then the solution blows up in finite time for any power of nonlinearity. This gives an affirmative answer for the conjecture that the critical exponent agrees with that of the wave equation when the damping is non-effective in one space dimension.

math.AP↗

arXiv2013

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.

math.AP↗

arXiv2012

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.

math.AP↗

arXiv2012

Small data blow-up of L^{2}-solution for the nonlinear Schrödinger equation without gauge invariance

We study the initial value problem for the nonlinear Schrödinger equation. We will prove that the blow-up of the L^{2}-norm of solutions with suitable initial data. We impose a condition related to the sign of the data but put no restriction on their size.

math.AP↗

arXiv2012

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

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 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.

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