This paper is concerned with the blow-up of solutions to the initial boundary value problem for the wave equation with scale invariant damping term on the exterior domain, where the nonlinear terms are power nonlinearity $|u|^{p} $
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u
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p
, derivative nonlinearity $|u_{t}|^{p} $
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u
t
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p
and combined nonlinearities $|u_{t}|^{p}+ |u|^{q} $
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u
t
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p
+
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u
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q
, respectively. Upper bound lifespan estimates of solutions to the problem are obtained by constructing suitable test functions and utilizing the test function technique. The main novelty is that lifespan estimates of solutions are associated with the well-known Strauss exponent and Glassey exponent. To the best of our knowledge, the results in Theorems 1.1–1.3 are new.
<abstract><p>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) = \varphi_{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\rightarrow 1^{-} $. To the best of our knowledge, the results in Theorems 1.1–1.3 are new.</p></abstract>
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