arXiv subjects
Publications and source records attributed to Motohiro Sobajima.
We provide a probabilistic characterization of criticality, subcriticality, and supercriticality for subordinated Schrödinger operators. We also investigate the relationship between the subcriticality of these operators and the uniform boundedness of solutions to the associated wave equation.
This paper is concerned with a threshold phenomenon for the existence of scattering states for nonlinear Schrödinger equations. The nonlinearity includes a non-oscillatory term of the order lower than the Strauss exponent. We show that no scattering states exist for the equation in a weighted Sobolev space. It is emphasized that our method admits initial data with good properties, such as compactly supported smooth functions. The result indicates that the Strauss exponent acts as a threshold for the power of the nonlinearity that determines whether solutions scatter or not in the weighted space.
In this paper, we give an alternative perspective of the criticality theory for (nonnegative) Schrödinger operators. Schrödinger operator $S=-Δ+V$ is classified as subcritical/critical in terms of the existence/nonexistence of a positive Green function for the associated elliptic equation $Su=f$. Such a property strongly affects to the large-time behavior of solutions to the parabolic equation $\partial_tv+Sv=0$. In this paper, we propose a remarkable quantity in terms of the structure of Hilbert lattices, which keeps some important properties including the notion of criticality theory. As an application, we study the large-time behavior of solutions to the hyperbolic equation $\partial_t^2w+Sw=0$.
The paper concerns with the decay property of solutions to the initial-boundary value problem of the semilinear heat equation $\partial_tu-Δu+u^p=0$ in exterior domains $Ω$ in $\mathbb{R}^N$ ($N\geq 2$). The problem for the one-dimensional case is formulated with $Ω=(0,\infty)$ which is one of the representative of the connected components in $\mathbb{R}$. One can see that the $C_0$-semigroup for the corresponding linear problem possesses an invariant measure $ϕ(x)\,dx$, where $ϕ$ is a positive harmonic function satisfying the Dirichlet boundary condition. This paper clarifies that the mass of solutions with respect to the measure $ϕ(x)\,dx$ vanishes as $t\to \infty$ if and only if $1 \min\{2,1+\frac{2}{N}\}$, we prove that all solutions are asymptotically free. The asymptotic profile is actually given by a modification with Gaussian when $N\geq 3$.
This paper is concerned with the existence/nonexistence of nontrivial global-in-time solutions to the Cauchy problem \begin{equation} \begin{cases}\tag{P}\partial_tu-\partial_x^2u+Vu=(1+x^2)^{-\frac{m}{2}}u^p,&x\in\mathbb{R},\ t>0,\\ u(x,0)=u_0(x)\ge0,&x\in\mathbb{R}, \end{cases} \end{equation} where $p>1$, $m\ge0$, $u_0\in BC(\mathbb{R})$ and the potential $V=V(x)\in BC(\mathbb{R})$ satisfies a certain property. More precisely, we determine the critical Fujita exponent for (P), that is, the threshold for the global existence/nonexistence of (P).
In this paper, blowup phenomenon for the semilinear wave equation with time-dependent speed of propagation and scattering damping is considered under the smallness of initial data. Our result contains small data blowup for sub-Strauss exponent for the simplest semilinear wave equation and also the one for semilinear generalized Tricomi equation. Key ingredient is so-called test function method (developed in Ikeda--Sobajima--Wakasa [10]) with a certain conservative quantity via a special solution with the Liouville--Green (or WKB) approximation.
In this paper, we mainly discuss asymptotic profiles of solutions to a class of abstract second-order evolution equations of the form $u''+Au+u'=0$ in real Hilbert spaces, where $A$ is a nonnegative selfadjoint operator. The main result is the asymptotic expansion for all initial data belonging to the energy space, which is naturally expected. This is an improvement for the previous work (required a sufficient regularity). As an application, we focus our attention to damped wave equations in an exterior domain in $\mathbb{R^N (N \geq 2)$ with the Dirichlet boundary condition. By using the asymptotic expansion in the present paper, we could derive the optimal decay rates of energy functional of solutions to damped wave equations. Moreover, some decay estimates of the local energy (energy functional restricted in a compact subset) can be observed via the asymptotic expansion.
We characterize the domain of the Schrödinger operators $S=-Δ+c|x|^{-α}$ in $L^p(\mathbb{R}^N)$, with $0<α<2$ and $c\in\mathbb{R}$. When $αp< N$, the domain characterization is essentially known and can be proved using different tools, for instance kernel estimates and potentials in the Kato class or in the reverse Hölder class. However,the other cases seem not to be known, so far.In this paper, we give the explicit description of the domain of $S$ for all range of parameters $p,α$ and $c$.
The optimality of decay properties of the one-dimensional damped wave equations with potentials belonging to a certain class is discussed. The typical ingredient is a variant of Nash inequality which involves an invariant measure for the corresponding Schrödinger semigroup. This enables us to find a sharp decay estimate from above. Moreover, the use of a test function method with the Nash-type inequality provides the decay estimate from below. The diffusion phenomena for the damped wave equations with potentials are also considered.
Lifespan estimates for semilinear damped wave equations of the form $\partial_t^2u-Δu+\partial_tu=|u|^p$ in a two dimensional exterior domain endowed with the Dirichlet boundary condition are dealt with. For the critical case of the semilinear heat equation $\partial_tv-Δv=v^2$ with the Dirichlet boundary condition and the initial condition $v(0)=\varepsilon f$, the corresponding lifespan can be estimated from below and above by $\exp(\exp(C\varepsilon^{-1}))$ with different constants $C$. This paper clarifies that the same estimates hold even for the critical semilinear damped wave equation in the exterior of the unit ball under the restriction of radial symmetry. To achieve this result, a new technique to control $L^1$-type norm and a new Gagliardo--Nirenberg type estimate with logarithmic weight are introduced.
We study the large time behavior of solutions to the wave equation with space-dependent damping in an exterior domain. We show that if the damping is effective, then the solution is asymptotically expanded in terms of solutions of corresponding parabolic equations. The main idea to obtain the asymptotic expansion is the decomposition of the solution of the damped wave equation into the solution of the corresponding parabolic problem and the time derivative of the solution of the damped wave equation with certain inhomogeneous term and initial data. The estimate of the remainder term is an application of weighted energy method with suitable supersolutions of the corresponding parabolic problem.
The global existence for semilinear wave equations with space-dependent critical damping $\partial_t^2u-Δu+\frac{V_0}{|x|}\partial_t u=f(u)$ in an exterior domain is dealt with, where $f(u)=|u|^{p-1}u$ and $f(u)=|u|^p$ are in mind. Existence and non-existence of global-in-time solutions are discussed. To obtain global existence, a weighted energy estimate for the linear problem is crucial. The proof of such a weighted energy estimate contains an alternative proof of energy estimates established by Ikehata--Todorova--Yordanov [J.\ Math.\ Soc.\ Japan (2013), 183--236] but this clarifies the precise independence of the location of the support of initial data. The blowup phenomena is verified by using a test function method with positive harmonic functions satisfying the Dirichlet boundary condition.
We study a kind of nonlinear wave equations with damping and potential, whose coefficients are both critical in the sense of the scaling and depend only on the spatial variables. Based on the earlier works, one may think there are two kinds of blow-up phenomenons when the exponent of the nonlinear term is small. It also means there are two kinds of law to determine the critical exponent. In this paper, we obtain a blow-up result and get the estimate of the upper bound of the lifespan in critical and sub-critical cases. All of the results support such a conjecture, although for now, the existence part is still open.
This paper is concerned with the upper bound of the lifespan of solutions to nonlinear Schrödinger equations with general homogeneous nonlinearity of the critical order. In [8], Masaki and the first author obtain the upper bound of the lifespan of solutions to our equation via a test function method introduced by [16, 17]. Their nonlinearity contains a non-oscillating term $|u|^{1+2/d}$ which causes difficultly for constructing an even small data global solution. The non-oscillating term corresponds to the $L^1$-scaling critical. In this paper, it turns out that the upper bound can be refined by employing an unified test function by Ikeda and the second author [5].
We consider the singular limit problem in a real Hilbert space for abstract second order evolution equations with a parameter $\varepsilon \in (0,1]$. We first give an alternative proof of the celebrated results due to Kisynski (1963) from the viewpoint of the energy method. Next we derive a more precise asymptotic profile as $\varepsilon \to +0$ of the solution itself depending on $\varepsilon$ under rather high regularity assumptions on the initial data.
The paper concerned with higher order asymptotic expansion of solutions to the Cauchy problem of abstract hyperbolic equations of the form $u''+Au+u'=0$ in a Hilbert space, where $A$ is a nonnegative selfadjoint operator. The result says that by assuming the regularity of initial data, asymptotic profiles (of arbitrary order) are explicitly written by using the semigroup $e^{-tA}$ generated by $-A$. To prove this, a kind of maximal regularity for $e^{-tA}$ is used.
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.
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.