Local well-posedness for the Cauchy problem of coupled system of generalized Camassa-Holm equations in the Besov spaces is established by employing the Littlewood-Paley theory and a priori estimate of solution to transport equation. Furthermore, the blow-up criterion of solutions to the problem is illustrated. Our main new contribution is that the effects of dissipative coefficient $\lambda$ and exponent $b$ in the nonlinear terms to the solutions are analyzed. To the best of our knowledge, the results in Theorems 1.1 and 1.2 are new.

This paper is concerned with blow-up results of solutions to coupled system of the Tricomi equations with derivative type nonlinearities. Upper bound lifespan estimates of solutions to the Cauchy problem with small initial values are derived by using the test function method (see the proof of Theorem 1.1) and iteration argument (see the proof of Theorem 1.2), respectively. Our main new contribution is that lifespan estimates of solutions to the problem in the sub-critical and critical cases which are connected with the Glassey conjecture are established. To the best knowledge of authors, the results in Theorems 1.1 and 1.2 are new.

The main purpose of this paper is to study behaviors of solutions to the Cauchy problem for coupled systems of semilinear damped wave equations with power nonlinearities on the Heisenberg group, where the Fujita critical exponent is determined. By means of the Bihari inequality as well as the Gagliardo-Nirenberg-type inequality and the contraction mapping principle, the local well-posedness of the Cauchy problem is successfully proved. Combining energy estimates and decay estimates together with a contradiction argument, the global existence of solutions is also demonstrated. At the same time, the blow-up results and upper bound estimation of the solutions is derived by using the test function technique. Our main new contribution is the derivation of a fundamental inequality utilized in weighted energy estimates, as well as the selection of exponents in the scaling of two convex functions for constructing test functions.

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 $k_1,k_2$.