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In this paper, we prove a blow-up result for a generalized semilinear Euler-Poisson-Darboux equation with polynomially growing speed of propagation, when the power of the semilinear term is a shift of the Strauss' exponent for the classical semilinear wave equation. Our proof is based on a comparison argument of Kato-type for a second-order ODE with time-dependent coefficients, an integral representation formula by Yagdjian and the Radon transform. As byproduct of our method, we derive upper bound estimates for the lifespan which coincide with the sharp one for the classical semilinear wave equation in the critical case.

In the present manuscript, we determine the critical condition for the nonlinearity in a semilinear wave equation with a derivative-type nonlinearity. More precisely, we consider a nonlinear term depending on the time derivative of the solution, which is the product of a power nonlinearity with critical Glassey exponent and a modulus of continuity. By employing Zhou's approach along a certain characteristic line, we prove the blow-up in finite time for classical solutions (under a suitable sign condition for the Cauchy data) and we derive upper bound estimates for the lifespan for a not Dini continuous modulus of continuity. Furthermore, in the 3-dimensional and radially symmetric case, by using weighted $L^{\infty}$ estimates, we establish the global existence of small data solutions for a Dini continuous modulus of continuity, and lower bound estimates for the lifespan in the not Dini continuous case. These results provide the regularity threshold (i.e. the Dini condition) for the modulus of continuity in the nonlinearity., Comment: This is a new version for the critical condition

We study the effect of a viscous dissipation on the Cauchy problem for a Cattaneo-type model in nonlinear acoustics, established by applying the Lighthill approximation for the viscous or inviscid fluid model. The contribution of this paper is twofold. For the nonlinear viscous Cattaneo-type model involving a fractional Laplacian $(-\Delta)^{\alpha}$ in the viscous damping with $\alpha\in[0,1]$, we derive optimal decay rates for global (in time) solutions with small data in certain Sobolev spaces. Furthermore, by introducing a threshold $\alpha=1/2$ for the power of the fractional viscous dissipation, we derive an anomalous diffusion profile when $\alpha\in[0,1/2)$ and a diffusion wave profile when $\alpha\in[1/2,1]$ for large-time. Whereas, for the nonlinear inviscid Cattaneo-type model (or the Jordan-Moore-Gibson-Thompson equation in the critical case), we obtain the blow-up of the energy solutions in finite time under suitable assumptions for the initial data. Thus, the presence of a viscous dissipation in the nonlinear Cattaneo-type model is a criterion for the global (in time) existence and blow-up of solutions., Comment: Blow-up for the inviscid model (or the JMGT equation in the critical case) is added

In this work we derive some blow-up results for semilinear wave equations both in de Sitter and anti-de Sitter spacetimes. By requiring suitable conditions on a time-dependent factor in the nonlinear term, we prove the blow-up in finite time of the spatial averages of local in time solutions. In particular, we derive a sequence of lower bound estimates for the spatial average by combining a suitable slicing procedure with an iteration frame for this time-dependent functional.

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 an ODI result 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.

In this paper, we consider a weakly coupled system of a wave and damped Klein-Gordon equation with nonlinearities of derivative type. We prove a blow-up result for the Cauchy problem associated with this system for nonnegative and compactly supported data by means of an iteration argument.

In this note, we derive a blow-up result for a semilinear generalized Tricomi equation with damping and mass terms having time-dependent coefficients. We consider these coefficients with critical decay rates. Due to this threshold nature of the time-dependent coefficients (both for the damping and for the mass), the multiplicative constants appearing in these lower-order terms strongly influence the value of the critical exponent, determining a competition between a Fujita-type exponent and a Strauss-type exponent.

We study a fundamental model in nonlinear acoustics, precisely, the general Blackstock's model (that is, without Becker's assumption) in the whole space $\mathbb{R}^n$. This model describes nonlinear acoustics in perfect gases under the irrotational flow. By means of the Fourier analysis we will derive $L^2$ estimates for the solution of the linear homogeneous problem and its derivatives. Then, we will apply these estimates to study three different topics: the optimality of the decay estimates in the case $n\geqslant 5$ and the optimal growth rate for the $L^2$-norm of the solution for $n=3,4$; the singular limit problem in determining the first- and second-order profiles for the solution of the linear Blackstock's model with respect to the small thermal diffusivity; the proof of the existence of global (in time) small data Sobolev solutions with suitable regularity for a nonlinear Blackstock's model., Comment: Indiana Univ. Math. J. (in press)

In this article, we investigate the blow-up for local solutions to a semilinear wave equation in the generalized Einstein - de Sitter spacetime with nonlinearity of derivative type. More precisely, we consider a semilinear damped wave equation with a time-dependent and not summable speed of propagation and with a time-dependent coefficient for the linear damping term with critical decay rate. We prove in this work that the results obtained in a previous work, where the damping coefficient takes two particular values $0$ or $2$, can be extended for any positive damping coefficient. In the blow-up case, the upper bound of the exponent of the nonlinear term is given, and the lifespan estimate of the global existence time is derived as well.

In this paper, we establish blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime with nonlinearity of derivative type. Our approach is based on the integral representation formula for the solution to the corresponding linear problem in the one-dimensional case, that we will determine through Yagdjian's Integral Transform approach. As upper bound for the exponent of the nonlinear term, we discover a Glassey-type exponent which depends both on the space dimension and on the Lorentzian metric in the generalized Einstein-de Sitter spacetime.

In the present paper, we investigate the blow-up dynamics for local solutions to the semilinear generalized Tricomi equation with combined nonlinearity. As a result, we enlarge the blow-up region in comparison to the ones for the corresponding semilinear models with either power nonlinearity or nonlinearity of derivative type. Our approach is based on an iteration argument to establish lower bound estimates for the space average of local solutions. Finally, we obtain upper bound estimates for the lifespan of local solutions as byproduct of our iteration argument.

We prove by using an iteration argument some blow-up results for a semilinear damped wave equation in generalized Einstein-de Sitter spacetime with a time-dependent coefficient for the damping term and power nonlinearity. Then, we conjecture an expression for the critical exponent due to the main blow-up results, which is consistent with many special cases of the considered model and provides a natural generalization of Strauss exponent. In the critical case, we consider a non-autonomous and parameter-dependent Cauchy problem for a linear ODE of second-order, whose explicit solutions are determined by means of special functions' theory., Comment: arXiv admin note: text overlap with arXiv:2009.04388

In this paper, we prove some blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime by using an iteration argument and we derive upper bound estimates for the lifespan. In particular, we will focus on the critical cases which require the employment of a slicing procedure in the iterative mechanism. Furthermore, in order to deal with the main critical case, we will introduce a non-autonomous and parameter-dependent Cauchy problem for a linear ODE of second-order, whose explicit solution will be determined by applying the theory of special functions.

In this note, we prove a blow-up result for a semilinear generalized Tricomi equation with nonlinear term of derivative type, i.e., for the equation $\mathscr{T}_{\!\!\ell} u = |\partial_t u|^p$, where $ \mathscr{T}_{\!\!\ell} = \partial_t^2-t^{2\ell}\Delta$. Smooth solutions blow up in finite time for positive Cauchy data when the exponent $p$ of the nonlinear term is below $\frac{\mathscr{Q}}{\mathscr{Q}-2}$, where $\mathscr{Q}=(\ell+1)n+1$ is the quasi-homogeneous dimension of the generalized Tricomi operator $\mathscr{T}_{\!\!\ell}$. Furthermore, we get also an upper bound estimate for the lifespan.

In this paper, we study the semilinear wave equation with lower order terms (damping and mass) and with power type nonlinearity $|u|^p$ on compact Lie groups. We will prove the global in time existence of small data solutions in the evolution energy space without requiring any lower bounds for $p>1$. In our approach, we employ some results from Fourier analysis on compact Lie groups., Comment: arXiv admin note: text overlap with arXiv:2005.13479, arXiv:2005.14543

In this note, we study the semilinear wave equation with power nonlinearity $|u|^p$ on compact Lie groups. First, we prove a local in time existence result in the energy space via Fourier analysis on compact Lie groups. Then, we prove a blow-up result for the semilinear Cauchy problem for any $p>1$, under suitable sign assumptions for the initial data. Furthermore, sharp lifespan estimates for local (in time) solutions are derived., Comment: arXiv admin note: substantial text overlap with arXiv:2005.13479

In this note, we prove a blow-up result for the semilinear damped wave equation in a compact Lie group with power nonlinearity $|u|^p$ for any $p>1$, under suitable integral sign assumptions for the initial data, by using an iteration argument. A byproduct of this method is the upper bound estimate for the lifespan of a local in time solution. As a preliminary result, a local (in time) existence result is proved in the energy space via Fourier analysis on compact Lie groups.

In this paper, we study the blow-up of solutions to the semilinear Moore-Gibson-Thompson (MGT) equation with nonlinearity of derivative type $|u_t|^p$ in the conservative case. We apply an iteration method in order to study both the subcritical case and the critical case. Hence, we obtain a blow-up result for the semilinear MGT equation (under suitable assumptions for initial data) when the exponent $p$ for the nonlinear term satisfies $11$ for $n=1$. In particular, we find the same blow-up range for $p$ as in the corresponding semilinear wave equation with nonlinearity of derivative type., Comment: arXiv admin note: text overlap with arXiv:1909.08838

In this work, the Cauchy problem for the semilinear Moore-Gibson-Thompson (MGT) equation with power nonlinearity $|u|^p$ on the right-hand side is studied. Applying $L^2-L^2$ estimates and a fixed point theorem, we obtain local (in time) existence of solutions to the semilinear MGT equation. Then, the blow-up of local in time solutions is proved by using an iteration method, under certain sign assumption for initial data, and providing that the exponent of the power of the nonlinearity fulfills $1 < p \leqslant p_{\mathrm{Str}}(n)$ for $n \geqslant2$ and $p>1$ for $n=1$. Here the Strauss exponent $p_{\mathrm{Str}}(n)$ is the critical exponent for the semilinear wave equation with power nonlinearity. In particular, in the limit case $p=p_{\mathrm{Str}}(n)$ a different approach with a weighted space average of a local in time solution is considered.

In this paper, we consider the Cauchy problem for the semilinear damped wave equation on the Heisenberg group with power nonlinearity. We prove that the critical exponent is the Fujita exponent $p_{\mathrm{Fuj}}(\mathscr{Q}) = 1+2 / \mathscr{Q}$, where $\mathscr{Q}$ is the homogeneous dimension of the Heisenberg group. On the one hand, we will prove the global existence of small data solutions for $p >p_{\mathrm{Fuj}}(\mathscr{Q})$ in an exponential weighted energy space. On the other hand, a blow-up result for $1 < p \leq p_{\mathrm{Fuj}}(\mathscr{Q})$ under certain integral sign assumptions for the Cauchy data by using the test function method.

This paper is devoted to the derivation of $L^2$ - $L^2$ decay estimates for the solution of the homogeneous linear damped wave equation on the Heisenberg group $\mathbf{H}_n$, for its time derivative and for its horizontal gradient. Moreover, we consider the improvement of these estimates when further $L^1(\mathbf{H}_n)$ regularity is required for the Cauchy data. Our approach will rely strongly on the group Fourier transform of $\mathbf{H}_n$ and on the properties of the Hermite functions that form a maximal orthonormal system for $L^2(\mathbb{R}^n)$ of eigenfunctions of the harmonic oscillator.

In this note, we prove blow-up results for semilinear wave models with damping and mass in the scale-invariant case and with nonlinear terms of derivative type. We consider the single equation and the weakly coupled system. In the first case we get a blow-up result for exponents below a certain shift of the Glassey exponent. For the weakly coupled system we find as critical curve a shift of the corresponding curve for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. Our approach follows the one for the respective classical wave equation by Zhou Yi. In particular, an explicit integral representation formula for a solution of the corresponding linear scale-invariant wave equation, which is derived by using Yagdjian's integral transform approach, is employed in the blow-up argument. While in the case of the single equation we may use a comparison argument, for the weakly coupled system an iteration argument is applied.

In this paper we consider the semilinear Cauchy problem for the heat equation with power nonlinearity in the Heisenberg group $\mathbf{H}_n$. The heat operator is given in this case by $\partial_t-\Delta_H$, where $\Delta_H$ is the so-called sub-Laplacian on $\mathbf{H}_n$. We prove that the Fujita exponent $1 + 2/Q$ is critical, where $Q = 2n + 2$ is the homogeneous dimension of $\mathbf{H}_n$. Furthermore, we prove sharp lifespan estimates for local in time solutions in the subcritical case and in the critical case. In order to get the upper bound estimate for the lifespan (especially, in the critical case) we employ a revisited test function method developed recently by Ikeda-Sobajima. On the other hand, to find the lower bound estimate for the lifespan we prove a local in time result in weighted $L^\infty$ space.

This paper is devoted to derive integral representation formulae for the solution of an inhomogeneous linear wave equation with time-dependent damping and mass terms, that are scale-invariant with respect to the so-called hyperbolic scaling. Yagdjian's integral transform approach is employed for this purpose. The main step in our argument consists in determining the kernel functions for the different integral terms, which are related to the source term and to initial data. We will start with the one dimensional case (in space). We point out that we may not apply in a straightforward way Duhamel's principle to deal with the source term since the coefficients of lower order terms make our model not invariant by time translation. On the contrary, we shall begin with the representation formula for the inhomogeneous equation with vanishing data by using a revised Duhamel's principle. Then, we will derive the representation of the solution in the homogeneous case with nontrivial data. After deriving the formula in the one dimensional case, the classical approach by spherical means is used in order to deal with the odd dimensional case. Finally, using the method of descent, the representation formula in the even dimensional case is proved.

In this paper we consider the blow-up of solutions to a weakly coupled system of semilinear damped wave equations in the scattering case with nonlinearities of mixed type, namely, in one equation a power nonlinearity and in the other a semilinear term of derivative type. The proof of the blow-up results is based on an iteration argument. As expected, due to the assumptions on the coefficients of the damping terms, we find as critical curve in the p-q plane for the pair of exponents (p,q) in the nonlinear terms the same one found by Hidano-Yokoyama and, recently, by Ikeda-Sobajima-Wakasa for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. In the critical and not-damped case we provide a different approach from the test function method applied by Ikeda-Sobajima-Wakasa to prove the blow-up of the solution on the critical curve, improving in some cases the upper bound estimate for the lifespan. More precisely, we combine an iteration argument with the so-called slicing method to show the blow-up dynamic of a weighted version of the functionals used in the subcritical case.

In this paper we consider the blow-up for solutions to a weakly coupled system of semilinear damped wave equations of derivative type in the scattering case. After introducing suitable functionals proposed by Lai-Takamura for the corresponding single semilinear equation, we employ Kato's lemma to derive the blow-up result in the subcritical case. On the other hand, in the critical case an iteration procedure based on the slicing method is employed. Let us point out that we find as critical curve in the p-q plane for the pair of exponents (p, q) in the nonlinear terms the same one as for the weakly coupled system of semilinear not-damped wave equations with the same kind of nonlinearities.

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.

In this note two blow-up results are proved for a weakly coupled system of semilinear wave equations with distinct scale-invariant lower order terms both in the subcritical case and in the critical case, when the damping and the mass terms make both equations in some sense "wave-like". In the proof of the subcritical case an iteration argument is used. This approach is based on a coupled system of nonlinear ordinary integral inequalities and lower bound estimates for the spatial integral of the nonlinearities. In the critical case we employ a test function type method, that has been developed recently by Ikeda-Sobajima-Wakasa and relies strongly on a family of certain self-similar solutions of the adjoint linear equation. Therefore, as critical curve in the p - q plane of the exponents of the power nonlinearities for this weakly coupled system we conjecture a shift of the critical curve for the corresponding weakly coupled system of semilinear wave equations.

In this work we determine the critical exponent for a weakly coupled system of semilinear wave equations with distinct scale-invariant lower order terms, when these terms make both equations in some sense parabolic-like. For the blow-up result the test functions method is applied, while for the global existence (in time) results we use L2-L2 estimates with additional L1 regularity., Comment: 19 pages

In this paper, we study the blow-up of solutions for semilinear wave equations with scale-invariant dissipation and mass in the case in which the model is somehow 'wave-like'. A Strauss type critical exponent is determined as the upper bound for the exponent in the nonlinearity in the main theorems. Two blow-up results are obtained for the sub-critical case and for the critical case, respectively. In both cases, an upper bound lifespan estimate is given., Comment: 23 pages

In the present article a semilinear wave equation with scale-invariant damping and mass is considered. The global (in time) existence of radial symmetric solutions in even spatial dimension $n$ is proved using weighted $L^\infty-L^\infty$ estimates, under the assumption that the multiplicative constants, which appear in the coefficients of damping and of mass terms, fulfill an interplay condition which yields somehow a "wave-like" model. In particular, combining this existence result with a recently proved blow-up result, a suitable shift of Strauss exponent is proved to be the critical exponent for the considered model. Moreover, the still open part of a conjecture done by D'Abbicco - Lucente - Reissig is proved to be true in the massless case., Comment: 28 pages

In this paper, we obtain a blow-up result for solutions to a semi-linear wave equation with scale-invariant dissipation and mass and power non-linearity, in the case in which the model has a "wave like" behavior. In order to achieve this goal, we perform a change of variables that transforms our starting equation in a strictly hyperbolic semi-linear wave equation with time-dependent speed of propagation. Then, we apply Kato's lemma to find a blow-up result for solutions to the transformed equation under some support and sign assumptions on the initial data. A special emphasis is placed on the limit case, that is, when the exponent p is exactly equal to the upper bound of the range of admissible values of p for which this blow-up result is valid. In this critical case an explicit integral representation formula for solutions of the corresponding linear Cauchy problem in 1d is derived. Finally, carrying out the inverse change of variables we get a non-existence result for global (in time) solutions to the original model., Comment: 33 pages

In this paper we consider the following Cauchy problem for the semi-linear wave equation with scale-invariant dissipation and mass and power non-linearity: \begin{align}\label{CP abstract} \begin{cases} u_{tt}-\Delta u+\dfrac{\mu_1}{1+t} u_t+\dfrac{\mu_2^2}{(1+t)^2}u=|u|^p, \\ u(0,x)=u_0(x), \,\, u_t(0,x)=u_1(x), \end{cases}\tag{$\star$} \end{align} where $\mu_1, \mu_2^2$ are nonnegative constants and $p>1$. On the one hand we will prove a global (in time) existence result for \eqref{CP abstract} under suitable assumptions on the coefficients $\mu_1, \mu_2^2$ of the damping and the mass term and on the exponent $p$, assuming the smallness of data in exponentially weighted energy spaces. On the other hand a blow-up result for \eqref{CP abstract} is proved for values of $p$ below a certain threshold, provided that the data satisfy some integral sign conditions. Combining these results we find the critical exponent for \eqref{CP abstract} in all space dimensions under certain assumptions on $\mu_1$ and $\mu_2^2$. Moreover, since the global existence result is based on a contradiction argument, it will be shown firstly a local (in time) existence result.