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For small-amplitude semilinear wave equations with power type nonlinearity on the first-order spatial derivative, the expected sharp upper bound on the lifespan of solutions is obtained for both critical cases and subcritical cases, for all spatial dimensions $ n>1 $. It is achieved uniformly by constructing the integral equations, deriving the ordinary differential inequality system, and iteration argument. Combined with the former works, the sharp lifespan estimates for this problem are completely established, at least for the spherical symmetric case. [ABSTRACT FROM AUTHOR]

In this paper, we show the so-called 'combined effect' of two different kinds of nonlinear terms for semilinear wave equations in one space dimension. This effect means that the lifespan, the maximal existence time, of the classical solution is shorter than the minimum of ones for each term. Such a special phenomenon has been observed in all space dimensions except for one space dimension. We succeed to pick up the combined effect in one space dimension by classifying the lifespan estimates according to whether the total integral of the initial speed is zero, or not, and it is obtained only in the first case. It is also remarkable that, including the combined effect, our results on the lifespan estimates are partially better than those of the general theory for nonlinear wave equations which was considered completed about 30 years ago. [ABSTRACT FROM AUTHOR]

This paper studies the upper bound of the lifespan of classical solutions of the initial value problems for one dimensional wave equations with quasilinear terms of space-, or time-derivatives of the unknown function. The result for the space-derivative case guarantees the optimality of the general theory for nonlinear wave equations, and its proof is carried out by combination of ordinary differential inequality and iteration method on the lower bound of the weighted functional of the solution. [ABSTRACT FROM AUTHOR]

In the present paper, we prove the blow-up in finite time for local solutions of a semilinear Cauchy problem associated with a wave equation in anti-de Sitter spacetime in the critical case. According to this purpose, we combine a result for ordinary differential inequalities with an iteration argument by using an explicit integral representation formula for the solution to a linear Cauchy problem associated with the wave equation in anti-de Sitter spacetime in one space dimension. [ABSTRACT FROM AUTHOR]

In this work we study the blow-up of solutions of a weakly coupled system of damped semilinear wave equations in the scattering case with power nonlinearities. We apply an iteration method to study both the subcritical case and the critical case. In the subcritical case our approach is based on lower bounds for the space averages of the components of local solutions. In the critical case we use the slicing method and a couple of auxiliary functions, recently introduced by Wakasa–Yordanov, to modify the definition of the functionals with the introduction of weight terms. In particular, we find as critical curve for the pair (p , q) of the exponents in the nonlinear terms the same one as for the weakly coupled system of semilinear wave equations with power nonlinearities. [ABSTRACT FROM AUTHOR]

Abstract In our previous two works, we studied the blow-up and lifespan estimates for damped wave equations with a power nonlinearity of the solution or its derivative, with scattering damping independently. In this work, we are devoted to establishing a similar result for a combined nonlinearity. Comparing to the result of wave equation without damping, one can say that the scattering damping has no influence. [ABSTRACT FROM AUTHOR]

It is well-known that the critical exponent for semilinear damped wave equations is Fujita exponent when the damping is effective. Lai, Takamura and Wakasa in 2017 have obtained a blow-up result not only for super-Fujita exponent but also for the one closely related to Strauss exponent when the damping is scaling invariant and its constant is relatively small, which has been recently extended by Ikeda and Sobajima. Introducing a multiplier for the time-derivative of the spatial integral of unknown functions, we succeed in employing the techniques on the analysis for semilinear wave equations and proving a blow-up result for semilinear damped wave equations with sub-Strauss exponent when the damping is in the scattering range. [ABSTRACT FROM AUTHOR]

We are interested in almost global existence cases in the general theory for nonlinear wave equations, which are caused by critical exponents of nonlinear terms. Such situations can be found in only three cases in the theory, cubic terms in two space dimensions, quadratic terms in three space dimensions and quadratic terms including a square of unknown functions itself in four space dimensions. Except for the last case, criteria to classify nonlinear terms into the almost global, or global existence case, are well-studied and known to be so-called null condition and non-positive condition. Our motivation of this work is to find such a kind of the criterion in four space dimensions. In our previous paper, an example of the non-single term for the almost global existence case is introduced. In this paper, we show an example of the global existence case. These two examples have nonlinear integral terms which are closely related to derivative loss due to high dimensions. But it may help us to describe the final form of the criterion. [ABSTRACT FROM AUTHOR]

We are interested in the upper bound of the lifespan of solutions of semilinear wave equations. For the sub-critical case in high dimensions, it has been believed that the basic tools of its analysis are Kato’s lemma on ordinary differential inequalities and the rescaling argument in the functional method. But there is a small lack of delicate analysis and no published paper about this. Here we give a simple alternative proof by means of improved Kato’s lemma without any rescaling argument. [ABSTRACT FROM AUTHOR]

Abstract: The final open part of Straussʼ conjecture on semilinear wave equations was the blow-up theorem for the critical case in high dimensions. This problem was solved by Yordanov and Zhang (2006) , or Zhou (2007) independently. But the estimate for the lifespan, the maximal existence time, of solutions was not clarified in both papers. In this paper, we refine their theorems and introduce a new iteration argument to get the sharp upper bound of the lifespan. As a result, with the sharp lower bound by Li and Zhou (1995) , the lifespan of solutions of in with the initial data of a small parameter , compactly supported smooth functions f and g, has an estimate where c and C are positive constants depending only on f and g. This upper bound has been known to be the last open optimality of the general theory for fully nonlinear wave equations. [Copyright &y& Elsevier]

In this paper we study a global in time existence of classical solutions of semilinear systems of 3-dimensional wave equations. We see that one component of the global in time solution can be arbitrarily large if another component is small enough according to some balance of each amplitude. Also its sharpness is discussed. This is a specific nature of strongly coupled systems. [ABSTRACT FROM AUTHOR]

This paper studies initial value problems for semilinear wave equations with spatial weights in one space dimension. The lifespan estimates of classical solutions for compactly supported data are established in all the cases of polynomial weights. The results are classified into two cases according to the total integral of the initial speed. [ABSTRACT FROM AUTHOR]

We are interested in the so-called "combined effect" of two different kinds of nonlinear terms for semilinear wave equations in one space dimension. Recently, the first result with the same formulation as in the higher dimensional case has been obtained if and only if the total integral of the initial speed is zero, namely Huygens' principle holds. In this paper, we extend the nonlinear term to the general form including the product type. Such model equations are extremely meaningful only in one space dimension because the most cases in higher dimensions possess the global-in-time existence of a classical solution in the general theory for nonlinear wave equations. It is also remarkable that our results on the lifespan estimates are partially better than those of the general theory. This fact tells us that there is a possibility to improve the general theory which was expected complete more than 30 years ago. [ABSTRACT FROM AUTHOR]

In this work we consider several semilinear damped wave equations with "subcritical" nonlinearities, focusing on studying lifespan estimates for energy solutions. Our main concern is on equations with scale-invariant damping and mass. By imposing different assumptions on the initial data, we prove lifespan estimates from above, distinguishing between "wave-like" and "heat-like" behaviours. Furthermore, we conjecture logarithmic improvements for the estimates on the "transition surfaces" separating the two behaviours. As a direct consequence, we reorganize the blow-up results and lifespan estimates for the massless case, and we obtain in particular improved lifespan estimates for the one dimensional case, compared to the known results. We also study semilinear wave equations with scattering damping and negative mass term, finding that if the decay rate of the mass term equals to 2, the lifespan estimate coincides with the one in a special case of scale-invariant damped equation. The main tool employed in the proof is a Kato's type lemma, established by iteration argument. [ABSTRACT FROM AUTHOR]

In this paper, we study the initial value problem for semilinear wave equations with the time-dependent and scale-invariant damping in two dimensions. Similarly to the one dimensional case by Kato, Takamura and Wakasa in 2019, we obtain the lifespan estimates of the solution for a special constant in the damping term, which are classified by total integral of the sum of the initial position and speed. The key fact is that, only in two space dimensions, such a special constant in the damping term is a threshold between "wave-like" domain and "heat-like" domain. As a result, we obtain a new type of estimate especially for the critical exponent. [ABSTRACT FROM AUTHOR]