Strauss exponent for semilinear wave equations with scattering space dependent damping

Lai, N.; Tu, Ziheng

doi:10.1016/j.jmaa.2020.124189

Cited by 30 publications

(15 citation statements)

References 64 publications

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“…The damped semilinear wave equation

({, \begin{matrix} left \\ array u_{t t} - Δ u + \frac{μ}{{(1 + t)}^{β}} u_{t} = f (u, u_{t}), t > 0, x \in ℝ^{n}, \\ array (u, u_{t}) (x, 0) = (ε u_{0} (x), ε u_{1} (x)), x \in ℝ^{n} \end{matrix})

has been studied extensively in recent years, where

\frac{μ}{{false(1 + t false)}^{β}} u_{t} false(β \in ℝ false)

is the damping term (see detailed illustration in previous works 24–40 ). Lai and Tu 36 consider the blow‐up dynamic and lifespan estimate of solution to semilinear wave equation with

f false(u, u_{t} false) = false| u {false|}^{p}, false| u_{t} {false|}^{p}

and damping term

\frac{μ}{{false(1 + false| x false| false)}^{β}} u_{t} false(β > 2 false)

by using test function approach, respectively. Lai et al 41 study the upper bound lifespan estimate of solution to semilinear wave equation with scale invariant damping term and mass term by employing the Kato lemma and iteration argument.…”

Section: Introductionmentioning

confidence: 99%

“…Inspired by previous works, 21,36,42–45 our purpose is to investigate lifespan estimates of solutions to problem () with power nonlinearities | v | p , | u | q , derivative nonlinearities | v t | p , | u t | q , mixed nonlinearities | v | q , | u t | p , and combined nonlinearities

false| v_{t} {false|}^{p_{1}} + false| v {false|}^{q_{1}}, false| u_{t} {false|}^{p_{2}} + false| u {false|}^{q_{2}}

in the subcritical and critical cases. Due to the similarity of structure in the equations, we expect that lifespan estimate of solution to the MGT equation is the same as wave equation.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Blow‐up and lifespan estimates of solutions to the weakly coupled system of semilinear Moore–Gibson–Thompson equations

Ming

Yang

Fan

2021

Math Methods in App Sciences

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This work is devoted to investigating formation of singularities of solutions to the weakly coupled system of semilinear Moore–Gibson–Thompson equations with power nonlinearities |v|p, |u|q, derivative nonlinearities |vt|p, |ut|q, mixed nonlinearities |v|q, |ut|p, and combined nonlinearities false|vtfalse|p1+false|vfalse|q1,0.1emfalse|utfalse|p2+false|ufalse|q2, respectively. Upper bound lifespan estimates of solutions to the problem in the subcritical and critical cases are also established. The main tool employed in the proofs is test function technique. The novelty is that lifespan estimates of solutions are connected with the well‐known Strauss exponent and Glassey exponent.

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“…The damped semilinear wave equation

({, \begin{matrix} left \\ array u_{t t} - Δ u + \frac{μ}{{(1 + t)}^{β}} u_{t} = f (u, u_{t}), t > 0, x \in ℝ^{n}, \\ array (u, u_{t}) (x, 0) = (ε u_{0} (x), ε u_{1} (x)), x \in ℝ^{n} \end{matrix})

has been studied extensively in recent years, where

\frac{μ}{{false(1 + t false)}^{β}} u_{t} false(β \in ℝ false)

is the damping term (see detailed illustration in previous works 24–40 ). Lai and Tu 36 consider the blow‐up dynamic and lifespan estimate of solution to semilinear wave equation with

f false(u, u_{t} false) = false| u {false|}^{p}, false| u_{t} {false|}^{p}

and damping term

\frac{μ}{{false(1 + false| x false| false)}^{β}} u_{t} false(β > 2 false)

Section: Introductionmentioning

confidence: 99%

false| v_{t} {false|}^{p_{1}} + false| v {false|}^{q_{1}}, false| u_{t} {false|}^{p_{2}} + false| u {false|}^{q_{2}}

in the subcritical and critical cases. Due to the similarity of structure in the equations, we expect that lifespan estimate of solution to the MGT equation is the same as wave equation.…”

Section: Introductionmentioning

confidence: 99%

Blow‐up and lifespan estimates of solutions to the weakly coupled system of semilinear Moore–Gibson–Thompson equations

Ming

Yang

Fan

2021

Math Methods in App Sciences

View full text Add to dashboard Cite

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“…The critical exponent for the case α < 0 is still given by p = 1 + 2 N −α in Nishihara-Sobajima-Wakasugi [22]. On the other hand, if α > 1, then Li-Tu [19] proved that the critical exponent for (1.4) is given by p = p S (N ). This means that the case α > 1 is close to the problem (1.2).…”

Section: Introductionmentioning

confidence: 99%

On global existence for semilinear wave equations with spacedependent critical damping

Sobajima¹

2021

Preprint

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The global existence for semilinear wave equations with space-dependent critical dampingin an exterior domain is dealt with, where f (u) = |u| p−1 u and f (u) = |u| p are in mind. Existence and non-existence of global-in-time solutions are discussed. To obtain global existence, a weighted energy estimate for the linear problem is crucial. The proof of such a weighted energy estimate contains an alternative proof of energy estimates established by Ikehata-Todorova-Yordanov [J. Math. Soc. Japan (2013), 183-236] but this clarifies the precise independence of the location of the support of initial data. The blowup phenomena is verified by using a test function method with positive harmonic functions satisfying the Dirichlet boundary condition.

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“…has been studied for a long time, where damping term satisfies g( [7,8,14,15,16,19]). Utilizing rescaled test function technique and iteration method, Ming et al [19] illustrate blow-up dynamic and lifespan estimate of solution to quasilinear wave equation with scattering damping µ (1+t) β u t (β > 1) and divergence form nonlinearity in the sub-critical and critical cases.…”

mentioning

confidence: 99%

“…Inspired by the works in [3,4,7,8,9,16,20,24], our interest is to establish blow-up results of solutions to problems (1.1) and (1.2). It is worth to mention that Dao et al [7,8] establish formation of singularity of solution to the semilinear structurally damped wave equation with |u| p and |u t | p , respectively.…”

mentioning

confidence: 99%

Formation of singularities of solutions to the Cauchy problem for semilinear Moore-Gibson-Thompson equations

Ming

Yang

Fan

2022

CPAA

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<p style='text-indent:20px;'>This paper is devoted to investigating formation of singularities for solutions to semilinear Moore-Gibson-Thompson equations with power type nonlinearity <inline-formula><tex-math id="M1">\begin{document}$ |u|^{p} $\end{document}</tex-math></inline-formula>, derivative type nonlinearity <inline-formula><tex-math id="M2">\begin{document}$ |u_{t}|^{p} $\end{document}</tex-math></inline-formula> and combined type nonlinearities <inline-formula><tex-math id="M3">\begin{document}$ |u_{t}|^{p}+|u|^{q} $\end{document}</tex-math></inline-formula> in the case of single equation, combined type nonlinearities <inline-formula><tex-math id="M4">\begin{document}$ |v_{t}|^{p_{1}}+|v|^{q_{1}} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ |u_{t}|^{p_{2}}+|u|^{q_{2}} $\end{document}</tex-math></inline-formula>, combined and power type nonlinearities <inline-formula><tex-math id="M6">\begin{document}$ |v_{t}|^{p_{1}}+|v|^{q_{1}} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ |u|^{q_{2}} $\end{document}</tex-math></inline-formula>, combined and derivative type nonlinearities <inline-formula><tex-math id="M8">\begin{document}$ |v_{t}|^{p_{1}}+|v|^{q_{1}} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ |u_{t}|^{p_{2}} $\end{document}</tex-math></inline-formula> in the case of coupled system, respectively. More precisely, blow-up results of solutions to problems in the sub-critical and critical cases are derived by applying test function technique. Moreover, upper bound lifespan estimates of solutions to the coupled systems are investigated. The main new contribution is that lifespan estimates of solutions are associated with the well-known Strauss exponent and Glassey exponent.</p>

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Strauss exponent for semilinear wave equations with scattering space dependent damping

Cited by 30 publications

References 64 publications

Blow‐up and lifespan estimates of solutions to the weakly coupled system of semilinear Moore–Gibson–Thompson equations

Blow‐up and lifespan estimates of solutions to the weakly coupled system of semilinear Moore–Gibson–Thompson equations

On global existence for semilinear wave equations with spacedependent critical damping

Formation of singularities of solutions to the Cauchy problem for semilinear Moore-Gibson-Thompson equations

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