Finite time blow-up for an inhomogeneous strongly damped fourth-order wave equation with nonlinear memory
- Authors
-
-
- Keywords:
- strongly damped, fourth-order, wave equation, nonlinear memory, inhomogeneous term, finite time blow-up
- Abstract
-
This paper investigates the blow-up phenomena for an inhomogeneous, strongly damped fourth-order wave equation featuring a nonlinear memory term
$$
u_{tt}(t,x) -\triangle^{2} u(t,x)- \triangle u_{t}(t,x) = \frac{1}{\Gamma(1-\alpha)} \int_{0}^{t}(t-s)^{-\alpha}|u(s,x)|^{p}ds+ \omega(x)
$$ with initial conditions $$ (u(0,x),\partial_t u(0,x))=(u_0(x),u_1(x)) $$ in \( \mathbb{R}^N ,\) where \( N\ge 1 \), \( p>1 \), \( \alpha\in(0,1) \), \( u_j\in L^1_{\mathrm{loc}}(\mathbb{R}^N) \) for \( j=0,1 \), and \( \omega(x)\not\equiv 0 \). By employing a special class of test functions, fractional calculus techniques, and nonlinear inequality methods, we prove that, assuming \( \omega\in L^1(\mathbb{R}^N) \) and \( \int_{\mathbb{R}^N}\omega(x)\,dx>0 \), the problem admits no global weak solution for any \( p>1 \). - References
-
[1] W. Chen and A. Z. Fino, Blow-up of the solutions to semilinear strongly damped wave equations with different nonlinear terms in an exterior domain, Mathematical Methods in the Applied Sciences 44 (2021), 6787–6807.
[2] H. Fujita, On the blowing-up of solutions of the Cauchy problem for u_t = Δu + u^{1+α}, Journal of the Faculty of Science, University of Tokyo, Sect. I A, Mathematics 13 (1966), no. 2, 109–124.
[3] J. B. Hartle and S. W. Hawking, Wave function of the universe, Physical Review D 28 (1983), no. 12, 2960–2975.
[4] M. Jleli and B. Samet, New critical behaviors for semilinear wave equations and systems with linear damping terms, Mathematical Methods in the Applied Sciences (2020), 1–11, Published online 27 Mar 2020.
[5] M. Jleli, B. Samet, and C. Vetro, Large time behavior for inhomogeneous damped wave equations with nonlinear memory, Symmetry 12 (2020), no. 10, 1609.
[6] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam, 2006.
[7] M. V. Klibanov, Global convexity in a three-dimensional inverse acoustic problem, SIAM Journal on Mathematical Analysis 28 (1997), no. 6, 1371–1388.
[8] V. Pata and S. Zelik, Smooth attractors for strongly damped wave equations, Nonlinearity 19 (2006), no. 7, 1495–1506.
[9] S. I. Pokhozhaev, Nonexistence of global solutions of nonlinear evolution equations, Differential Equations 49 (2013), no. 5, 599–606.
[10] G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, Journal of Differential Equations 174 (2001), no. 2, 464–489.
[11] X. Wang, R. Xu, and Y. Yang, Long-time behavior for fourth order nonlinear wave equations with dissipative and dispersive terms, Applied Numerical Mathematics 199 (2024), 248–265.
[12] Y. Yang and Z.-B. Fang, On a strongly damped semilinear wave equation with time-varying source and singular dissipation, Advances in Nonlinear Analysis 12 (2023), no. 1, 20220267.
- Additional Files
- Published
- 26-12-2025
- Issue
- Vol. 3 No. 1 (2025)
- Section
- Research Article
- License
-
Copyright (c) 2025 ChangYong Xiang
This work is licensed under a Creative Commons Attribution 4.0 International License.
How to Cite
Similar Articles
- Donal O'Regan, On nonlinear Sobolev equations with terminal observations in \(L^p\) spaces , Letters on Applied and Pure Mathematics: Vol. 1 No. 1 (2023)
- Caibin Zeng, Time fractional stochastic Navier-Stokes equations driven by fractional Brownian motion , Letters on Applied and Pure Mathematics: Vol. 2 No. 1 (2024)
- Yusuf Gurefe, On conformable differential equation with variable coefficient , Letters on Applied and Pure Mathematics: Vol. 3 No. 1 (2025)
- Vo Ngoc Minh, The continuity of solution set of a multi-point boundary problem with a control system , Letters on Applied and Pure Mathematics: Vol. 1 No. 1 (2023)
- Le Dinh Long, Ngo Ngoc Hung, Recovering initial condition backward problem for composite fractional relaxation equations , Letters on Applied and Pure Mathematics: Vol. 3 No. 1 (2025)
- Thái Anh Nhan, Relja Vulanovic, Parameter-uniform convergence analysis on a Bakhvalov-type mesh with a smooth mesh-generating function using the preconditioning approach , Letters on Applied and Pure Mathematics: Vol. 2 No. 1 (2024)
- Devendra Kumar, Jagdev Singh, On an ill-posed problem for system of coupled sinh-Gordon equations , Letters on Applied and Pure Mathematics: Vol. 1 No. 1 (2023)
- Vinh Q. Mai, Dat T. Nguyen, A mathematical model of enzymatic reactions with Bi-Bi random mechanism and its applications , Letters on Applied and Pure Mathematics: Vol. 1 No. 1 (2023)
- Mengjiao Feng, Junyilang Zhao, Existence and stability of periodic solutions in a multi-species GA predator-prey system with multiple delays on time scales , Letters on Applied and Pure Mathematics: Vol. 3 No. 1 (2025)
- Devendra Kumar, On approximation and error estimate for Poisson equation in Banach space , Letters on Applied and Pure Mathematics: Vol. 2 No. 1 (2024)
You may also start an advanced similarity search for this article.