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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 methods with suitable supersolutions of the corresponding parabolic problem. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic expansion of solutions to the initial-value problem of u ″ (t) + A u (t) + b (t) u ′ (t) = 0 in a Hilbert space with a nonnegative selfadjoint operator A and a coefficient b (t) ∼ (1 + t) − β (− 1 < β < 1). In the case b (t) ≡ 1 , it is known that the higher-order asymptotic profiles are determined via a family of first-order differential equations of the form v ′ (t) + A v (t) = F n (t) (Sobajima (2021) [10]). For the time-dependent case, it is only known that the asymptotic behavior of such a solution is given by the one of b (t) v ′ (t) + A v (t) = 0. The result of this paper is to find the equations for all higher-order asymptotic profiles. It is worth noticing that the equation for n -th order profile u ˜ n is given via v ′ (t) + m n (t) A v (t) = F n (t) which coefficient m n (time-scale) differs each other. [ABSTRACT FROM AUTHOR]

In this paper we consider the wave equations with power type nonlinearities including time-derivatives of unknown functions and their weakly coupled systems. We propose a framework of test function methods and give a simple proof of the derivation of sharp upper bounds for lifespan of solutions to nonlinear wave equations and their systems. We point out that for respective critical cases, we use a family of self-similar solutions to the linear wave equation including Gauss's hypergeometric functions, which are originally introduced by Zhou [59]. We emphasize that our framework does not require the pointwise positivity of the initial data even in the high dimensional case N ≥ 4. Moreover, we find a new (p , q) -curve for the system ∂ t 2 u − Δ u = | v | q , ∂ t 2 v − Δ v = | ∂ t u | p with lifespan estimates for small solutions in a new region. [ABSTRACT FROM AUTHOR]

We consider the initial value problem for a semi-linear wave equation with a time-dependent effective damping term. The interest is the behavior of lifespan of solutions in view of the asymptotic profile of the coefficient of damping as time tends to infinity. A simple case of effective damping and a threshold case in the sense of overdamping were discussed in Ikeda and Wakasugi (2015) and Ikeda and Inui (2019), respectively. We discuss here general damping terms under a certain assumption that is discussed. The result of this paper is the sharp lifespan estimates of blowup solutions including the typical cases treated in previous studies. The proof of the upper bound of lifespan is a modification of the test-function method by Ikeda–Sobajima (2019) and the one of lower bound is based on the technique of scaling variables introduced by Gallay and Raugel (1998) for the one-dimensional case and Wakasugi (2017) for the higher-dimensional case. [ABSTRACT FROM AUTHOR]

This paper is concerned with weighted energy estimates for solutions to wave equation ∂ t 2 u − Δ u + a (x) ∂ t u = 0 with space-dependent damping term a (x) = | x | − α (α ∈ [ 0 , 1 ]) in an exterior domain Ω having a smooth boundary. The main result asserts that the weighted energy estimates with weight function like polynomials are given and these decay rates are almost sharp, even when the initial data do not have compact support in Ω. The crucial idea is to use special solution of ∂ t u = | x | α Δ u including Kummer's confluent hypergeometric functions. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the blowup phenomena for initial–boundary value problem (0.1) τ ∂ t 2 u (x , t) − Δ u (x , t) + a (x) ∂ t u (x , t) = λ | u (x , t) | p , (x , t) ∈ C Σ × (0 , T) , u (x , t) = 0 , (x , t) ∈ ∂ C Σ × (0 , T) , u (x , 0) = ε f (x) , x ∈ C Σ , τ ∂ t u (x , 0) = τ ε g (x) , x ∈ C Σ , where C Σ is a cone-like domain in R N (N ≥ 2) defined as C Σ = int r ω ∈ R N ; r ≥ 0 , ω ∈ Σ with a connected open set Σ in S N − 1 with smooth boundary ∂ Σ. If N = 1 , then we only consider two cases C Σ = (0 , ∞) and C Σ = R. Here a (x) is a non-zero coefficient of ∂ t u which could be complex-valued and space-dependent, λ ∈ ℂ is a fixed constant, and ε > 0 is a small parameter. The constants τ = 0 , 1 switch the parabolicity and hyperbolicity of the problem (0.1). The result proposes an argument for sharp upper lifespan estimates of solutions to (0.1) by a test function method based on Mitidieri and Pokhozhaev (2001). The crucial idea is to introduce an ordinary differential inequality with a parameter as a variable. [ABSTRACT FROM AUTHOR]

Abstract In this paper we consider the following initial-boundary value problem with the power type nonlinearity | u | p with 1 < p ≤ 2 in a two-dimensional exterior domain (0.1) { τ ∂ t 2 u (x , t) − Δ u (x , t) + e i ζ ∂ t u (x , t) = λ | u (x , t) | p , (x , t) ∈ Ω × (0 , T) , u (x , t) = 0 , (x , t) ∈ ∂ Ω × (0 , T) , u (x , 0) = ε f (x) , x ∈ Ω , ∂ t u (x , 0) = ε g (x) , x ∈ Ω , where Ω is given by Ω = { x ∈ R 2 ; | x | > 1 } , ζ ∈ [ − π 2 , π 2 ] , λ ∈ C and τ ∈ { 0 , 1 } switches the parabolicity, dispersivity and hyperbolicity. Remark that 2 = 1 + 2 / N is well-known as the Fujita exponent. If p > 2 , then there exists a small global-in-time solution of (0.1) for some initial data small enough (see Ikehata [11]), and if p < 2 , then global-in-time solutions cannot exist for any positive initial data (see Ogawa–Takeda [22] and Lai–Yin [14]). The result is that for given initial data (f , τ g) ∈ H 0 1 (Ω) × L 2 (Ω) satisfying (f + τ g) log ⁡ | x | ∈ L 1 (Ω) with some requirement, the solution blows up at finite time, and moreover, the upper bound for lifespan of solutions to (0.1) is given as the following double exponential type when p = 2 : LifeSpan (u) ≤ exp ⁡ [ exp ⁡ (C ε − 1) ]. The crucial idea is to use test functions which approximates the harmonic function log ⁡ | x | satisfying Dirichlet boundary condition and the technique modified from [9]. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We prove that, for ... the operator ... admits realizations generating analytic semigroups in Lp(RN). Moreover, for α < p-N/p, we provide an explicit description of the domain. [ABSTRACT FROM AUTHOR]

We show an abstract method to construct different realizations of abstract operators generating a strongly continuous semigroup and apply it to several second order elliptic operators with singular coefficients. [ABSTRACT FROM AUTHOR]

Hardy inequalities for smooth functions vanishing at a negligible set incontained in 0 is considered. The result gives best constants of these inequalities, which depends on the structure of the corresponding negligible set. As applications, related Schrödinger operators are also considered. [ABSTRACT FROM AUTHOR]

Abstract: Generation of analytic semigroups on Banach space X by is shown, where A is a negative generator of a bounded analytic semigroup on X, B is a closed operator in X belonging to a class related to A and is a parameter satisfying for some . The proof may be regarded as a modification of the perturbation argument established by Okazawa [8]. As applications, the generation of non-contractive analytic semigroups by Schrödinger operators with inverse square potentials in is discussed. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the solvability for the Cauchy problem of heat equations in ( ) with singular lower order terms: where , , is an initial value, f is an inhomogeneous term and is arbitrarily given. In particular, and f are taken from a weighted -space. The result generalizes the one of Baras and Goldstein (1984) [2] from to and improves the one of Arendt, Goldstein and Goldstein (2006) [1] which deals with an m-accretive realization of the linear part in a weighted -space under the stronger condition . Moreover, this paper clarifies some regularities of solutions. [Copyright &y& Elsevier]

Abstract: Generation of analytic semigroups by generalized Ornstein–Uhlenbeck operators in the weighted space is shown. The results improve the recent results established by Metafune–Prüss–Rhandi–Schnaubelt (2005) [8] ( ) and Kojima–Yokota (2010) [6] ( ). The proofs of those are based on a generation result acting on the unweighted space [8, Theorem 3.4], while ours is carried out by an application of -theory for second-order elliptic operators [13, Theorem 1.3] obtained by Sobajima (2012). Consequently, the additional condition assumed in both [8,6] is completely removed. [Copyright &y& Elsevier]

We consider the initial-boundary value problem of semilinear wave equation with nonlinearity | u | p in exterior domain in R N (N ≥ 3). Especially, the lifespan of blowup solutions with small initial data are studied. The result gives upper bounds of lifespan which is essentially the same as the Cauchy problem in R N. At least in the case N = 4 , their estimates are sharp in view of the work by Zha–Zhou [21]. The idea of the proof is to use special solutions to linear wave equation with Dirichlet boundary condition which are constructed via an argument based on Wakasa–Yordanov [15]. [ABSTRACT FROM AUTHOR]

We prove precise - estimates for the semigroup generated by in . [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the Cauchy problem (CGL) in for complex Ginzburg–Landau equations with Laplacian Δ and nonlinear term multiplied by the complex coefficients and , respectively (, , , ). The global existence of strong solutions to (CGL) is established without any upper restriction on but with some restriction on and . The result corresponds to Ginibre and Velo (1996) which is technically proved by combining convolution (regularizing) methods with compactness (localizing) methods, while our proof here is fairly simplified. The key to our proof is the Cauchy problem (CGL) R which is (CGL) with Δ replaced with , where (), (). The solvability of (CGL) R is a direct consequence of Okazawa and Yokota (2002) . Taking the limit of global strong solutions to (CGL) R as yields a global strong solution to (CGL). The result gives also an unbounded version of Okazawa and Yokota (2002) for the initial–boundary value problem on bounded domains. [Copyright &y& Elsevier]

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. [ABSTRACT FROM AUTHOR]