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Yuta Wakasugi

Publications and source records attributed to Yuta Wakasugi.

At least 37 records · Page 2Linked to original sources
arXiv2020

Asymptotic behavior of solutions to elliptic and parabolic equations with unbounded coefficients of the second orderin unbounded domains

We study an asymptotic behavior of solutions to elliptic equations of the second order in a two dimensional exterior domain. Under the assumption that the solution belongs to $L^q$ with $q \in [2,\infty)$, we prove a pointwise asymptotic estimate of the solution at the spatial infinity in terms of the behavior of the coefficients. As a corollary, we obtain the Liouville-type theorem in the case when the coefficients may grow at the spacial infinity. We also study a corresponding parabolic problem in the $n$-dimensional whole space and discuss the energy identity for solutions in $L^q$. As a corollary we show also the Liouville-type theorem for both forward and ancient solutions.

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arXiv2020

Asymptotic properties of steady and nonsteady solutions to the 2D Navier-Stokes equations with finite generalized Dirichlet integral

We consider the stationary and non-stationary Navier-Stokes equations in the whole plane $\mathbb{R}^2$ and in the exterior domain outside of the large circle. The solution $v$ is handled in the class with $\nabla v \in L^q$ for $q \ge 2$. Since we deal with the case $q \ge 2$, our class is larger in the sense of spatial decay at infinity than that of the finite Dirichlet integral, i.e., for $q=2$ where a number of results such as asymptotic behavior of solutions have been observed. For the stationary problem we shall show that $ω(x)= o(|x|^{-(1/q + 1/q^2)})$ and $\nabla v(x) = o(|x|^{-(1/q+1/q^2)} \log |x|)$ as $|x| \to \infty$, where $ω\equiv {\rm rot\,} v$. As an application, we prove the Liouville type theorem under the assumption that $ω\in L^q(\mathbb{R}^2)$. For the non-stationary problem, a generalized $L^q$-energy identity is clarified. We also apply it to the uniqueness of the Cauchy problem and the Liouville type theorem for ancient solutions under the assumption that $ω\in L^q(\mathbb{R}^2 \times I)$.

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arXiv2020

The Cauchy problem of the semilinear second order evolution equation with fractional Laplacian and damping

In the present paper, we prove time decay estimates of solutions in weighted Sobolev spaces to the second order evolution equation with fractional Laplacian and damping for data in Besov spaces. Our estimates generalize the estimates obtained in the previous studies. The second aim of this article is to apply these estimates to prove small data global well-posedness for the Cauchy problem of the equation with power nonlinearities. Especially, the estimates obtained in this paper enable us to treat more general conditions on the nonlinearities and the spatial dimension than the results in the previous studies.

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arXiv2019

Supersolutions for parabolic equations with unbounded diffusion and its applications to some classes of parabolic and hyperbolic equations

This paper is concerned with supersolutions to parabolic equations of the form \begin{equation} \partial_t U (x,t)-D(x)ΔU(x,t)=0, \quad (x,t)\in \mathbb{R}^N \times (0,\infty), \end{equation} where $D\in C(\mathbb{R}^N)$ is positive. Under the behavior of the diffusion coefficient $D$ with polynomial order at spatial infinity, a family of supersolutions with slowly decaying property at spatial infinity is provided. As a first application, weighted $L^2$ type decay estimates for the initial-boundary value problem of the corresponding parabolic equation are proved. The second application is the study of the exterior problem of wave equations with space-dependent damping terms. By using supersolutions provided above, energy estimates with polynomial weight and diffusion phenomena are shown.

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arXiv2019

Endpoint Strichartz estimate for the damped wave equation and its application

Recently, the Strichartz estimates for the damped wave equation was obtained by the first author except for the wave endpoint case. In the present paper, we give the Strichartz estimate in the wave endpoint case. We slightly modify the argument of Keel--Tao. Moreover, we apply the endpoint Strichartz estimate to the unconditional uniqueness for the energy critical nonlinear damped wave equation. This problem seems not to be solvable as the perturbation of the wave equation.

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arXiv2019

Sharp lifespan estimates of blowup solutions to semilinear wave equations with time-dependent effective damping

We consider the initial value problem for the semilinear wave equation with time-dependent effective damping. The interest is the behavior of lifespan of solutions in view of the asymptotic profile of the damping as $t\to \infty$. The result of this paper is the sharp lifespan estimates of blowup solutions for general time-dependent damping including threshold cases between effective and overdamping.

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arXiv2018

Critical exponent for the semilinear wave equations with a damping increasing in the far field

We consider the Cauchy problem of the semilinear wave equation with a damping term \begin{align*} u_{tt} - Δu + c(t,x) u_t = |u|^p, \quad (t,x)\in (0,\infty)\times \mathbb{R}^N,\quad u(0,x) = \varepsilon u_0(x), \ u_t(0,x) = \varepsilon u_1(x), \quad x\in \mathbb{R}^N, \end{align*} where $p>1$ and the coefficient of the damping term has the form \begin{align*} c(t,x) = a_0 (1+|x|^2)^{-α/2} (1+t)^{-β} \end{align*} with some $a_0 > 0$, $α< 0$, $β\in (-1, 1]$. In particular, we mainly consider the cases $ α< 0, β=0$ or $α< 0, β= 1$, which imply $α+ β< 1$, namely, the damping is spatially increasing and effective. Our aim is to prove that the critical exponent is given by $ p = 1+ \frac{2}{N-α}$. This shows that the critical exponent is the same as that of the corresponding parabolic equation $c(t,x) v_t - Δv = |v|^p$. The global existence part is proved by a weighted energy estimates with an exponential-type weight function and a special case of the Caffarelli-Kohn-Nirenberg inequality. The blow-up part is proved by a test-function method introduced by Ikeda and Sobajima (arXiv:1710.06780v1). We also give an upper estimate of the lifespan.

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arXiv2018

Revisit on the blow-up rate of solutions for a weakly coupled system of semilinear heat equations in the subcritical case

We introduce a straightforward method to analyze the blow-up of systems of ordinary differential inequalities, and apply it to study the blow- up of solutions to a weakly coupled system of semilinear heat equations. We prove that the solution blows up in a finite time under the subcritical condition. Moreover, we give estimates of lifespan of the solution and lower estimates of the blow-up rate for a localized average of each component.

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arXiv2018

$L^p$-$L^q$ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data

We study the Cauchy problem of the damped wave equation \begin{align*} \partial_{t}^2 u - Δu + \partial_t u = 0 \end{align*} and give sharp $L^p$-$L^q$ estimates of the solution for $1\le q \le p < \infty\ (p\neq 1)$ with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial data in $(H^s\cap H_r^β) \times (H^{s-1} \cap L^r)$ with $r \in (1,2]$, $s\ge 0$, and $β= (n-1)|\frac{1}{2}-\frac{1}{r}|$, and prove the local and global existence of solutions. In particular, we prove the existence of the global solution with small initial data for the critical nonlinearity with the power $1+\frac{2r}{n}$, while it is known that the critical power $1+\frac{2}{n}$ belongs to the blow-up region when $r=1$. We also discuss the asymptotic behavior of the global solution in supercritical cases. Moreover, we present blow-up results in subcritical cases. We give estimates of lifespan and blow-up rates by an ODE argument.

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arXiv2018

Global well-posedness for the semilinear wave equation with time dependent damping in the overdamping case

We study global existence of solutions to the Cauchy problem for the wave equation with time-dependent damping and a power nonlinearity in the overdamping case. We prove the global well-posedness for small data in the energy space for the whole energy-subcritical case. This result implies that small data blow-up does not occur in the overdamping case, different from the other cases, i.e. effective or non-effective damping.

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arXiv2017

Finite energy of generalized suitable weak solutions to the Navier-Stokes equations and Liouville-type theorems in two dimensional domains

Introducing a new notion of generalized suitable weak solutions, we first prove validity of the energy inequality for such a class of weak solutions to the Navier-Stokes equations in the whole space $\mathbb{R}^n$. Although we need certain growth condition on the pressure, we may treat the class even with infinite energy quantity except for the initial velocity. We next handle the equation for vorticity in 2D unbounded domains. Under a certain condition on the asymptotic behavior at infinity, we prove that the vorticity and its gradient of solutions are both globally square integrable. As their applications, Loiuville-type theorems are obtained.

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arXiv2017

Blow-up of solutions for weakly coupled systems of complex Ginzburg-Landau equations

Blow-up phenomena ofvweakly coupled systems of several evolution equations, especially complex Ginzburg-Landau equationsvis shown by a straightforward ODE approach not so-called test-function method, which gives the natural blow-up rate. The difficulty of the proof is that, unlike the single case, terms which come from the fact that the Laplacian cannot be absorbed into the weakly coupled nonlinearities. A similar ODE approach is applied to heat systems by Mochizuki to obtain the lower estimate of lifespan.

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arXiv2017

Weighted energy estimates for wave equation with space-dependent damping term for slowly decaying initial data

This paper is concerned with weighted energy estimates for solutions to wave equation $\partial_t^2u-Δu + a(x)\partial_tu=0$ with space-dependent damping term $a(x)=|x|^{-α}$ $(α\in [0,1))$ in an exterior domain $Ω$ having a smooth boundary. The main result asserts that the weighted energy estimates with weight function like polymonials are given and these decay rate are almost sharp, even when the initial data do not have compact support in $Ω$. The crucial idea is to use special solution of $\partial_t u=|x|^αΔu$ including Kummer's confluent hypergeometric functions.

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arXiv2017

Diffusion phenomena for the wave equation with space-dependent damping term growing at infinity

In this paper, we study the asymptotic behavior of solutions to the wave equation with damping depending on the space variable and growing at the spatial infinity. We prove that the solution is approximated by that of the corresponding heat equation as time tends to infinity. The proof is based on semigroup estimates for the corresponding heat equation and weighted energy estimates for the damped wave equation. To construct a suitable weight function for the energy estimates, we study a certain elliptic problem.

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arXiv2017

Movement of time-delayed hot spots in Euclidean space for special initial states

We consider the Cauchy problem for the damped wave equation under the initial state that the sum of an initial position and an initial velocity vanishes. When the initial position is non-zero, non-negative and compactly supported, we study the large time behavior of the spatial null, critical, maximum and minimum sets of the solution. The spatial null set becomes a smooth hyper-surface homeomorphic to a sphere after a large enough time. The spatial critical set has at least three points after a large enough time. The set of spatial maximum points escapes from the convex hull of the support of the initial position. The set of spatial minimum points consists of one point after a large time, and the unique spatial minimum point converges to the centroid of the initial position at time infinity.

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arXiv2016

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 diffusion 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 coefficient of the damping term is radially symmetric. In this paper, by modifying their elliptic problem, we establish weighted energy estimates and diffusion phenomena even when the coefficient of the damping term is not radially symmetric.

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