We consider the initial value problem for the semilinear Schrödinger equation: (NLS) i∂tu + ∆u = λ|u| p , (t, x) ∈ [0, T) × R n , where p > 1, λ ∈ C\ {0}. In this paper, we will prove a small data blow-up result of L 2 and H 1-solution for (NLS) in 1 < p < 1 + 4/n. Also, an upper bound of the lifespan will be given (Theorem 2.2).

We consider the Cauchy problem for the semilinear Schrödinger equationis a C-valued given function and T λ is a maximal existence time of the solution. Our first aim in the present paper is to prove a large data blow-up result for (NLS) in H s -critical or H s -subcritical case p ≤ p s := 1 + 4/(d − 2s), for some s ≥ 0. More precisely, we show that in the case 1 < p ≤ p s , for a suitable H s -function f , there exists λ 0 > 0 such that for any λ > λ 0 , the following estimateshold, where κ, C > 0 are constants independent of λ and (r, ρ) is an admissible pair (see Theorem 2.3). Our second aim is to prove non-existence local weak-solution for (NLS) in the H s -supercritical case p > p s , which means that if p > p s , then there exists an H s -function f such that if there exist T > 0 and a weak-solution u to (NLS) on [0, T ), then λ = 0 (see Theorem 2.5).

We give a sufficient condition for global existence of the solutions to a generalized derivative nonlinear Schrödinger equation (gDNLS) by a variational argument. The variational argument is applicable to a cubic derivative nonlinear Schrödinger equation (DNLS). For (DNLS), Wu (2015) proved that the solution with the initial data u 0 is global if u 0 2 L 2 < 4π by the sharp Gagliardo-Nirenberg inequality. The variational argument gives us another proof of the global existence for (DNLS). Moreover, by the variational argument, we can show that the solution to (DNLS) is global if the initial data u 0 satisfies u 0 2 L 2 = 4π and the momentum P(u 0 ) is negative.

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.1 , min{2, n n−2 }] if 3 ≤ n ≤ 6. Finite time blow-up of local solutions was also obtained for any n ≥ 1 and 1 < p < 2r n . However, the above global well-posedness results are restricted to n ≤ 6 and there are no results for higher dimensional cases. Also, Narazaki [42] considered the slowly decaying data belonging to modulation spaces and proved the global existence when the nonlinearity has integer power.Concerning the asymptotic profile of global solutions, Gallay and Raugel [5] determined the asymptotic expansion up to the second order when n = 1 and the initial data belongs to the weighted Sobolev space H 1,1 × H 0,1 (see Section 1.2 for the definition). Using the expansion of solutions to the heat equation, Kawakami and Ueda [31] extended it to the case n ≤ 3. Hayashi, Kaikina and Naumkin [9] obtained the first order asymptotics for all n ≥ 1 and the initial data belonging to (H s,0 ∩ H 0,α ) × (H s−1,0 ∩ H 0,α ) with α > n 2 (particularly, belonging to L 1 ). Recently, Takeda [63, 64] determined the higher order asymptotic expansion of global solutions. Narazaki and Nishihara [43] studied the case of slowly decaying data and proved that if n ≤ 3 and the data behaves like (1 + |x|) −kn with 0 < k ≤ 1, then, the asymptotic profile of the global solution is given by G(t, x) * (1 + |x|) −kn , where G is the Gaussian and * denotes the convolution with respect to spatial variables.Related to the equation (1.1), systems of nonlinear damped wave equation were studied and the critical exponent and the asymptotic behavior of solutions were investigated (see [61,41,62,53,42,54,49,50,15,51]).In the present paper, we establish the large data local well-posedness and the small data global well-posedness for the nonlinear damped wave equation (1.1) with slowly decaying initial data. Our global well-posedness results extend those of [25,43] to all space dimensions, and generalize that of [9] to slowly decaying initial data. Moreover, we study the asymptotic profile of the global solution. This also extends those of [43] to all space dimensions. Considering the asymptotic behavior of solutions in weighted norms, we further extended the result of [9] to the asymptotics in L m -norm with m ≤ 2. Finally, we give an almost optimal lifespan estimate from both above and below. This is also an extension of [36,47,20], in which L 1 -data were treated.1.1. Main results. We say that u ∈ L ∞ (0, T ; L 2 (R n )) is a mild solution of (1.1) if u satisfies the ...

We study the Cauchy problem of the damped wave equation ∂ 2 t u − ∆u + ∂tu = 0 and give sharp L p -L q estimates of the solution for 1 ≤ q ≤ p < ∞ (p = 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, s ≥ 0, and β = (n − 1)| 1 2 − 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 + 2r n , while it is known that the critical power 1+ 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 by an ODE argument.

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We consider the following semirelativistic nonlinear Schrödinger equation (SNLS): { i ∂ t u ± ( m 2 − Δ ) 1 / 2 u = λ | u | p , a m p ; ( t , x ) ∈ [ 0 , T ) × R d , u ( 0 , x ) = u 0 ( x ) , a m p ; x ∈ R d , \begin{equation} \left \{ \begin {array}{ll} i\partial _t u \pm (m^2-\Delta )^{1/2} u = \lambda |u|^{p}, & (t,x)\in [0,T)\times \mathbb {R}^d, \\ u(0,x)=u_0(x), & x \in \mathbb {R}^d, \end{array} \right . \notag \end{equation} where m ≥ 0 m\geq 0 , λ ∈ C ∖ { 0 } \lambda \in \mathbb {C} \setminus \{ 0\} , d ∈ N d\in \mathbb {N} , T > 0 T>0 , and ∂ t = ∂ / ∂ t \partial _t=\partial /\partial t . Here ( m 2 − Δ ) 1 / 2 := F − 1 ( m 2 + | ξ | 2 ) 1 / 2 F (m^2-\Delta )^{1/2}:=\mathcal {F}^{-1} (m^2+|\xi |^2 )^{1/2} \mathcal {F} , where F \mathcal {F} denotes the Fourier transform. Fujiwara and Ozawa proved the nonexistence of global weak solutions to SNLS for some initial data in the case of d = 1 d=1 , m = 0 m=0 , and 1 > p ≤ 2 1>p\leq 2 by a test function method. In this paper, we extend their result to a more general setting: for example, m ≥ 0 m\geq 0 , d ∈ N d\in \mathbb {N} , or p > 1 p>1 . Moreover, we obtain the upper estimates of weak solutions to SNLS. The key to the proof is to choose an appropriate test function.

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