This article is mainly concerned with the formation of singularity for a solution to the Cauchy problem of the semilinear Moore-Gibson-Thompson equation with general initial values and different types of nonlinear memory terms $N_{\gamma ,q}(u)$, $N_{\gamma ,p}(u_t)$, $N_{\gamma ,p,q}(u,u_t)$. The proof of the blow-up phenomenon for the solution in the whole space is based on the test function method ( $\psi (x,t)={\phi }_R(x)D_{t|T}^{\alpha }(w(t))$). It is worth pointing out that the Moore-Gibson-Thompson equation with memory terms can be regarded as an approximation of the nonlinear Moore-Gibson-Thompson equation when $\gamma \to 1^{-}$. To the best of our knowledge, the results in Theorems 1.1–1.3 are new.

This paper is mainly concerned with the initial boundary value problems of semilinear wave equations with damping term and mass term as well as Neumann boundary conditions on exterior domain in three dimensions. Blow-up and upper bound lifespan estimates of solutions to the problem with damping term and mass term are derived by applying test function technique and iterative method, where nonlinear terms are power nonlinearity $|u{|}^p$, derivative nonlinearity $|u_t{|}^p$, combined nonlinearities $|u_t{|}^p+|u{|}^q$, respectively. Moreover, upper bound lifespan estimate of solution to the problem with scale invariant damping term, non-negative mass term and combined nonlinearities $|u_t{|}^p+|u{|}^q$ is obtained. The proofs are based on the test function method and iterative approach. The main new contribution is that upper bound lifespan estimates of solutions are associated with the Strauss exponent and Glassey exponent. In addition, the variation trend of wave is achieved by taking advantage of numerical simulation.

This work was concerned with the weakly coupled system of semi-linear wave equations with time dependent speeds of propagation, damping terms, and derivative nonlinear terms in generalized Einstein-de Sitter space-time on ${\mathbb{R}}^n$. Under certain assumptions about the indexes $k_1,k_2$, coefficients ${\mu }_1,{\mu }_2$, and nonlinearity exponents $p,q$, applying the iteration technique, finite time blow-up of local solutions to the small initial value problem of the coupled system was investigated. Blow-up region and upper bound lifespan estimate of solutions to the problem were established. Compared with blow-up results in the previous literature, the new ingredient relied on that the blow-up region of solutions obtained in this work varies due to the influence of coefficients $k_1,k_2$.