# American Institute of Mathematical Sciences

doi: 10.3934/eect.2022006
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## Blow-up of solutions to semilinear wave equations with a time-dependent strong damping

 1 Department of Mathematics, Sultan Qaboos University, P.O. Box 46, Al-Khoud 123, Muscat, Oman 2 Basic Sciences Department, Deanship of Preparatory Year and Supporting Studies, P.O. Box 1982, Imam Abdulrahman Bin Faisal University, Dammam, KSA

* Corresponding author: Ahmad Z. Fino

Received  November 2021 Revised  December 2021 Early access February 2022

The paper investigates a class of a semilinear wave equation with time-dependent damping term ($-\frac{1}{{(1+t)}^{\beta}}\Delta u_t$) and a nonlinearity $|u|^p$. We will show the influence of the parameter $\beta$ in the blow-up results under some hypothesis on the initial data and the exponent $p$ by using the test function method. We also study the local existence in time of mild solution in the energy space $H^1(\mathbb{R}^n)\times L^2(\mathbb{R}^n)$.

Citation: Ahmad Z. Fino, Mohamed Ali Hamza. Blow-up of solutions to semilinear wave equations with a time-dependent strong damping. Evolution Equations and Control Theory, doi: 10.3934/eect.2022006
##### References:
 [1] T. Cazenave and A. Haraux, Introduction aux Problèemes D'évolution Semi-Linéaires, Ellipses, Paris, 1990. [2] M. D'Abbicco and M. Reissig, Semilinear structural damped waves, Math. Meth. Appl. Sci., 37 (2014), 1570-1592.  doi: 10.1002/mma.2913. [3] L. D'Ambrosio and S. Lucente, Nonlinear Liouville theorems for Grushin and Tricomi operators, J. Differential Equations, 193 (2003), 511-541.  doi: 10.1016/S0022-0396(03)00138-4. [4] A. Z. Fino, Finite time blow up for wave equations with strong damping in an exterior domain, Mediterr. J. Math., 17 (2020), Paper No. 174, 21 pp. doi: 10.1007/s00009-020-01607-2. [5] A. Fino and G. Karch, Decay of mass for nonlinear equation with fractional Laplacian, Monatsh. Math., 160 (2010), 375-384.  doi: 10.1007/s00605-009-0093-3. [6] A. Z. Fino and M. Kirane, Qualitative properties of solutions to a time-space fractional evolution equation, Quart. Appl. Math., 70 (2012), 133-157.  doi: 10.1090/S0033-569X-2011-01246-9. [7] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.  doi: 10.1007/BF01647974. [8] T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math., 33 (1980), 501-505.  doi: 10.1002/cpa.3160330403. [9] È. Mitidieri and S. I. Pohozaev, A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Proc. Steklov. Inst. Math., 234 (2001), 1-362. [10] A. Pazy, Semi-Groups of Linear Operators and Applications to Partial Differential Equations, , Appl. Math. Sci., 44. Springer, New-York, 1983. doi: 10.1007/978-1-4612-5561-1. [11] W. A. Strauss, Nonlinear scattering theory at low energy, J. Functional Analysis, 41 (1981), 110-133.  doi: 10.1016/0022-1236(81)90063-X. [12] Y. Wakasugi, On the Diffusive Structure for the Damped Wave Equation with Variable Coefficients, Ph. D thesis, Osaka University, 2014. [13] B. T. Yordanov and Q. S. Zhang, Finite time blow up for critical wave equations in high dimensions, J. Funct. Anal., 231 (2006), 361-374.  doi: 10.1016/j.jfa.2005.03.012. [14] Q. S. Zhang, A blow up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 109-114.  doi: 10.1016/S0764-4442(01)01999-1. [15] Y. Zhou, Blow up of solutions to semilinear wave equations with critical exponent in high dimensions, Chin. Ann. Math. Ser. B, 28 (2007), 205-212.  doi: 10.1007/s11401-005-0205-x.

show all references

##### References:
 [1] T. Cazenave and A. Haraux, Introduction aux Problèemes D'évolution Semi-Linéaires, Ellipses, Paris, 1990. [2] M. D'Abbicco and M. Reissig, Semilinear structural damped waves, Math. Meth. Appl. Sci., 37 (2014), 1570-1592.  doi: 10.1002/mma.2913. [3] L. D'Ambrosio and S. Lucente, Nonlinear Liouville theorems for Grushin and Tricomi operators, J. Differential Equations, 193 (2003), 511-541.  doi: 10.1016/S0022-0396(03)00138-4. [4] A. Z. Fino, Finite time blow up for wave equations with strong damping in an exterior domain, Mediterr. J. Math., 17 (2020), Paper No. 174, 21 pp. doi: 10.1007/s00009-020-01607-2. [5] A. Fino and G. Karch, Decay of mass for nonlinear equation with fractional Laplacian, Monatsh. Math., 160 (2010), 375-384.  doi: 10.1007/s00605-009-0093-3. [6] A. Z. Fino and M. Kirane, Qualitative properties of solutions to a time-space fractional evolution equation, Quart. Appl. Math., 70 (2012), 133-157.  doi: 10.1090/S0033-569X-2011-01246-9. [7] F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math., 28 (1979), 235-268.  doi: 10.1007/BF01647974. [8] T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math., 33 (1980), 501-505.  doi: 10.1002/cpa.3160330403. [9] È. Mitidieri and S. I. Pohozaev, A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities, Proc. Steklov. Inst. Math., 234 (2001), 1-362. [10] A. Pazy, Semi-Groups of Linear Operators and Applications to Partial Differential Equations, , Appl. Math. Sci., 44. Springer, New-York, 1983. doi: 10.1007/978-1-4612-5561-1. [11] W. A. Strauss, Nonlinear scattering theory at low energy, J. Functional Analysis, 41 (1981), 110-133.  doi: 10.1016/0022-1236(81)90063-X. [12] Y. Wakasugi, On the Diffusive Structure for the Damped Wave Equation with Variable Coefficients, Ph. D thesis, Osaka University, 2014. [13] B. T. Yordanov and Q. S. Zhang, Finite time blow up for critical wave equations in high dimensions, J. Funct. Anal., 231 (2006), 361-374.  doi: 10.1016/j.jfa.2005.03.012. [14] Q. S. Zhang, A blow up result for a nonlinear wave equation with damping: The critical case, C. R. Acad. Sci. Paris Sér. I Math., 333 (2001), 109-114.  doi: 10.1016/S0764-4442(01)01999-1. [15] Y. Zhou, Blow up of solutions to semilinear wave equations with critical exponent in high dimensions, Chin. Ann. Math. Ser. B, 28 (2007), 205-212.  doi: 10.1007/s11401-005-0205-x.
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