$ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data

Masahiro Ikeda; Takahisa Inui; Mamoru Okamoto; Yuta Wakasugi

doi:10.3934/cpaa.2019090

Previous Article
Existence, multiplicity and concentration for a class of fractional $ p \& q $ Laplacian problems in $ \mathbb{R} ^{N} $
CPAA Home
This Issue
Next Article
Spike layer solutions for a singularly perturbed Neumann problem: Variational construction without a nondegeneracy

July 2019, 18(4): 1967-2008. doi: 10.3934/cpaa.2019090

$ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data

Masahiro Ikeda ^1,2, , Takahisa Inui ^3, , Mamoru Okamoto ^4, and Yuta Wakasugi ^5,

1.	Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1, Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan
2.	Center for Advanced Intelligence Project, RIKEN, Japan
3.	Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
4.	Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano City 380-8553, Japan
5.	Department of Engineering for Production and Environment, Graduate School of Science and Engineering, Ehime University, 3 Bunkyo-cho, Matsuyama, Ehime, 790-8577, Japan

Received August 2018 Revised October 2018 Published January 2019

We study the Cauchy problem of the damped wave equation

$ \begin{align*} \partial_{t}^2 u - \Delta u + \partial_t u = 0 \end{align*} $

and give sharp

$ L^p $

$ L^q $

estimates of the solution for

$ 1\le q \le p < \infty\ (p\neq 1) $

with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial data in

$ (H^s\cap H_r^{\beta}) \times (H^{s-1} \cap L^r) $

with

$ r \in (1,2] $

$ s\ge 0 $

, and

$ \beta = (n-1)|\frac{1}{2}-\frac{1}{r}| $

, and prove the local and global existence of solutions. In particular, we prove the existence of the global solution with small initial data for the critical nonlinearity with the power

$ 1+\frac{2r}{n} $

, while it is known that the critical power

$ 1+\frac{2}{n} $

belongs to the blow-up region when

$ r = 1 $

. We also discuss the asymptotic behavior of the global solution in supercritical cases. Moreover, we present blow-up results in subcritical cases. We give estimates of lifespan by an ODE argument.

Keywords: Nonlinear damped wave equation, $ L^p $-$ L^q $ estimates, critical exponent.

Mathematics Subject Classification: Primary: 35L71; Secondary: 35A01, 35B40, 35B44.

Citation: Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta Wakasugi. $ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1967-2008. doi: 10.3934/cpaa.2019090

References:

[1]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. Google Scholar
[2]	J. Chen, D. Fan and C. Zhang, Space-time estimates on damped fractional wave equation, Abstr. Appl. Anal., (2014). doi: 10.1155/2014/428909. Google Scholar
[3]	J. Chen, D. Fan and C. Zhang, Estimates for damped fractional wave equations and applications, Electronic Journal of Differential Equations, 2015 (2015), 1-14. Google Scholar
[4]	F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C. Google Scholar
[5]	K. Fujiwara, M. Ikeda and Y. Wakasugi, Estimates of lifespan and blow-up rates for the wave equation with a time-dependent damping and a power-type nonlinearity, Funkcial. Ekvac., to appear, arXiv: 1609.01035v2. Google Scholar
[6]	K. Fujiwara and T. Ozawa, Finite time blowup of solutions to the nonlinear Schrödinger equation without gauge invariance, J. Math. Phys., 57 (2016), 1-8. doi: 10.1063/1.4960725. Google Scholar
[7]	M.-H. Giga, Y. Giga and J. Saal, Nonlinear Partial Differential Equations, Progress in Nonlinear Differential Equations and their Applications, 79, Birkhäuser, Boston, MA, 2010. doi: 10.1007/978-0-8176-4651-6. Google Scholar
[8]	L. Grafakos, Classical Fourier Analysis, Third edition, Graduate Texts in Mathematics, 249, Springer, New York, 2014. ⅹⅷ+638 pp. doi: 10.1007/978-1-4939-1194-3. Google Scholar
[9]	N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with super critical nonlinearities, Differential Integral Equations, 17 (2004), 637-652. Google Scholar
[10]	N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with a critical nonlinearity, Trans. Amer. Math. Soc., 358 (2006), 1165-1185. doi: 10.1090/S0002-9947-05-03818-3. Google Scholar
[11]	N. Hayashi, E. I. Kaikina and P. I. Naumkin, On the critical nonlinear damped wave equation with large initial data, J. Math. Anal. Appl., 334 (2007), 1400-1425. doi: 10.1016/j.jmaa.2007.01.021. Google Scholar
[12]	N. Hayashi and P. I. Naumkin, Damped wave equation with a critical nonlinearity in higher space dimensions, J. Math. Appl. Anal., 446 (2017), 801-822. doi: 10.1016/j.jmaa.2016.09.005. Google Scholar
[13]	T. Hosono and T. Ogawa, Large time behavior and $L^p$-$L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differential Equations, 203 (2004), 82-118. doi: 10.1016/j.jde.2004.03.034. Google Scholar
[14]	M. Ikeda, T. Inui and Y. Wakasugi, The Cauchy problem for the nonlinear damped wave equation with slowly decaying data, NoDEA Nonlinear Differential Equations Appl., 24 (2017), no. 2, Art. 10, 53 pp. doi: 10.1007/s00030-017-0434-1. Google Scholar
[15]	R. Ikehata, Y. Miyaoka and T. Nakatake, Decay estimates of solutions for dissipative wave equations in $\mathbf R^N$ with lower power nonlinearities, J. Math. Soc. Japan, 56 (2004), 365-373. Google Scholar
[16]	R. Ikehata, K. Nishihara and H. Zhao, Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption, J. Differential Equations, 226 (2006), 1-29. Google Scholar
[17]	R. Ikehata and M. Ohta, Critical exponents for semilinear dissipative wave equations in $\mathbf R^N$, J. Math. Anal. Appl., 269 (2002), 87-97. doi: 10.1016/S0022-247X(02)00021-5. Google Scholar
[18]	R. Ikehata and K. Tanizawa, Global existence of solutions for semilinear damped wave equations in $\mathbf R^N$ with noncompactly supported initial data, Nonlinear Anal., 61 (2005), 1189-1208. doi: 10.1016/j.na.2005.01.097. Google Scholar
[19]	G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math., 143 (2000), 175-197. doi: 10.4064/sm-143-2-175-197. Google Scholar
[20]	S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan, 47 (1995), 617-653. doi: 10.2969/jmsj/04740617. Google Scholar
[21]	M. Kirane and M. Qafsaoui, Fujita's exponent for a semilinear wave equation with linear damping, Adv. Nonlinear Stud., 2 (2002), 41-49. doi: 10.1515/ans-2002-0103. Google Scholar
[22]	T.-T. Li and Y. Zhou, Breakdown of solutions to $\square u+u_t=\|u\|^{1+\alpha}$, Discrete Contin. Dynam. Syst., 1 (1995), 503-520. doi: 10.3934/dcds.1995.1.503. Google Scholar
[23]	P. Marcati and K. Nishihara, The $L^p$-$L^q$ estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media, J. Differential Equations, 191 (2003), 445-469. doi: 10.1016/S0022-0396(03)00026-3. Google Scholar
[24]	A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., 12 (1976), 169-189. doi: 10.2977/prims/1195190962. Google Scholar
[25]	A. Miyachi, On some estimates for the wave equation in $L^p$ and $H^p$, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (1980), 331-354. Google Scholar
[26]	M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z., 214 (1993), 325-342. doi: 10.1007/BF02572407. Google Scholar
[27]	T. Narazaki, $L^p$-$L^q$ estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan, 56 (2004), 585-626. doi: 10.2969/jmsj/1191418647. Google Scholar
[28]	T. Narazaki, Global solutions to the Cauchy problem for a system of damped wave equations, Differential and Integral Equations, 24 (2011), 569-600. Google Scholar
[29]	T. Narazaki and K. Nishihara, Asymptotic behavior of solutions for the damped wave equation with slowly decaying data, J. Math. Anal. Appl., 338 (2008), 803-819. doi: 10.1016/j.jmaa.2007.05.068. Google Scholar
[30]	K. Nishihara, $L^p$-$L^q$ estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z., 244 (2003), 631-649. doi: 10.1007/s00209-003-0516-0. Google Scholar
[31]	J. C. Peral, $L^p$ estimates for the wave equation, J. Funct. Anal., 36 (1980), 114-145. doi: 10.1016/0022-1236(80)90110-X. Google Scholar
[32]	S. Sakata and Y. Wakasugi, Movement of time-delayed hot spots in Euclidean space, Math. Z., 285 (2017), 1007-1040. doi: 10.1007/s00209-016-1735-5. Google Scholar
[33]	S. Sjöstrand, On the Riesz means of the solutions of the Schrödinger equation, Ann. Scuola Norm. Sup. Pisa, 24 (1970), 331-348. Google Scholar
[34]	G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2001), 464-489. doi: 10.1006/jdeq.2000.3933. Google Scholar
[35]	F. B. Weissler, Existence and non-existence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845. Google Scholar
[36]	H. Yang and A. Milani, On the diffusion phenomenon of quasilinear hyperbolic waves, Bull. Sci. Math., 124 (2000), 415-433. doi: 10.1016/S0007-4497(00)00141-X. Google Scholar
[37]	Qi 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. Google Scholar

show all references

References:

[1]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. Google Scholar
[2]	J. Chen, D. Fan and C. Zhang, Space-time estimates on damped fractional wave equation, Abstr. Appl. Anal., (2014). doi: 10.1155/2014/428909. Google Scholar
[3]	J. Chen, D. Fan and C. Zhang, Estimates for damped fractional wave equations and applications, Electronic Journal of Differential Equations, 2015 (2015), 1-14. Google Scholar
[4]	F. M. Christ and M. I. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal., 100 (1991), 87-109. doi: 10.1016/0022-1236(91)90103-C. Google Scholar
[5]	K. Fujiwara, M. Ikeda and Y. Wakasugi, Estimates of lifespan and blow-up rates for the wave equation with a time-dependent damping and a power-type nonlinearity, Funkcial. Ekvac., to appear, arXiv: 1609.01035v2. Google Scholar
[6]	K. Fujiwara and T. Ozawa, Finite time blowup of solutions to the nonlinear Schrödinger equation without gauge invariance, J. Math. Phys., 57 (2016), 1-8. doi: 10.1063/1.4960725. Google Scholar
[7]	M.-H. Giga, Y. Giga and J. Saal, Nonlinear Partial Differential Equations, Progress in Nonlinear Differential Equations and their Applications, 79, Birkhäuser, Boston, MA, 2010. doi: 10.1007/978-0-8176-4651-6. Google Scholar
[8]	L. Grafakos, Classical Fourier Analysis, Third edition, Graduate Texts in Mathematics, 249, Springer, New York, 2014. ⅹⅷ+638 pp. doi: 10.1007/978-1-4939-1194-3. Google Scholar
[9]	N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with super critical nonlinearities, Differential Integral Equations, 17 (2004), 637-652. Google Scholar
[10]	N. Hayashi, E. I. Kaikina and P. I. Naumkin, Damped wave equation with a critical nonlinearity, Trans. Amer. Math. Soc., 358 (2006), 1165-1185. doi: 10.1090/S0002-9947-05-03818-3. Google Scholar
[11]	N. Hayashi, E. I. Kaikina and P. I. Naumkin, On the critical nonlinear damped wave equation with large initial data, J. Math. Anal. Appl., 334 (2007), 1400-1425. doi: 10.1016/j.jmaa.2007.01.021. Google Scholar
[12]	N. Hayashi and P. I. Naumkin, Damped wave equation with a critical nonlinearity in higher space dimensions, J. Math. Appl. Anal., 446 (2017), 801-822. doi: 10.1016/j.jmaa.2016.09.005. Google Scholar
[13]	T. Hosono and T. Ogawa, Large time behavior and $L^p$-$L^q$ estimate of solutions of 2-dimensional nonlinear damped wave equations, J. Differential Equations, 203 (2004), 82-118. doi: 10.1016/j.jde.2004.03.034. Google Scholar
[14]	M. Ikeda, T. Inui and Y. Wakasugi, The Cauchy problem for the nonlinear damped wave equation with slowly decaying data, NoDEA Nonlinear Differential Equations Appl., 24 (2017), no. 2, Art. 10, 53 pp. doi: 10.1007/s00030-017-0434-1. Google Scholar
[15]	R. Ikehata, Y. Miyaoka and T. Nakatake, Decay estimates of solutions for dissipative wave equations in $\mathbf R^N$ with lower power nonlinearities, J. Math. Soc. Japan, 56 (2004), 365-373. Google Scholar
[16]	R. Ikehata, K. Nishihara and H. Zhao, Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption, J. Differential Equations, 226 (2006), 1-29. Google Scholar
[17]	R. Ikehata and M. Ohta, Critical exponents for semilinear dissipative wave equations in $\mathbf R^N$, J. Math. Anal. Appl., 269 (2002), 87-97. doi: 10.1016/S0022-247X(02)00021-5. Google Scholar
[18]	R. Ikehata and K. Tanizawa, Global existence of solutions for semilinear damped wave equations in $\mathbf R^N$ with noncompactly supported initial data, Nonlinear Anal., 61 (2005), 1189-1208. doi: 10.1016/j.na.2005.01.097. Google Scholar
[19]	G. Karch, Selfsimilar profiles in large time asymptotics of solutions to damped wave equations, Studia Math., 143 (2000), 175-197. doi: 10.4064/sm-143-2-175-197. Google Scholar
[20]	S. Kawashima, M. Nakao and K. Ono, On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term, J. Math. Soc. Japan, 47 (1995), 617-653. doi: 10.2969/jmsj/04740617. Google Scholar
[21]	M. Kirane and M. Qafsaoui, Fujita's exponent for a semilinear wave equation with linear damping, Adv. Nonlinear Stud., 2 (2002), 41-49. doi: 10.1515/ans-2002-0103. Google Scholar
[22]	T.-T. Li and Y. Zhou, Breakdown of solutions to $\square u+u_t=\|u\|^{1+\alpha}$, Discrete Contin. Dynam. Syst., 1 (1995), 503-520. doi: 10.3934/dcds.1995.1.503. Google Scholar
[23]	P. Marcati and K. Nishihara, The $L^p$-$L^q$ estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media, J. Differential Equations, 191 (2003), 445-469. doi: 10.1016/S0022-0396(03)00026-3. Google Scholar
[24]	A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. Res. Inst. Math. Sci., 12 (1976), 169-189. doi: 10.2977/prims/1195190962. Google Scholar
[25]	A. Miyachi, On some estimates for the wave equation in $L^p$ and $H^p$, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (1980), 331-354. Google Scholar
[26]	M. Nakao and K. Ono, Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z., 214 (1993), 325-342. doi: 10.1007/BF02572407. Google Scholar
[27]	T. Narazaki, $L^p$-$L^q$ estimates for damped wave equations and their applications to semi-linear problem, J. Math. Soc. Japan, 56 (2004), 585-626. doi: 10.2969/jmsj/1191418647. Google Scholar
[28]	T. Narazaki, Global solutions to the Cauchy problem for a system of damped wave equations, Differential and Integral Equations, 24 (2011), 569-600. Google Scholar
[29]	T. Narazaki and K. Nishihara, Asymptotic behavior of solutions for the damped wave equation with slowly decaying data, J. Math. Anal. Appl., 338 (2008), 803-819. doi: 10.1016/j.jmaa.2007.05.068. Google Scholar
[30]	K. Nishihara, $L^p$-$L^q$ estimates of solutions to the damped wave equation in 3-dimensional space and their application, Math. Z., 244 (2003), 631-649. doi: 10.1007/s00209-003-0516-0. Google Scholar
[31]	J. C. Peral, $L^p$ estimates for the wave equation, J. Funct. Anal., 36 (1980), 114-145. doi: 10.1016/0022-1236(80)90110-X. Google Scholar
[32]	S. Sakata and Y. Wakasugi, Movement of time-delayed hot spots in Euclidean space, Math. Z., 285 (2017), 1007-1040. doi: 10.1007/s00209-016-1735-5. Google Scholar
[33]	S. Sjöstrand, On the Riesz means of the solutions of the Schrödinger equation, Ann. Scuola Norm. Sup. Pisa, 24 (1970), 331-348. Google Scholar
[34]	G. Todorova and B. Yordanov, Critical exponent for a nonlinear wave equation with damping, J. Differential Equations, 174 (2001), 464-489. doi: 10.1006/jdeq.2000.3933. Google Scholar
[35]	F. B. Weissler, Existence and non-existence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845. Google Scholar
[36]	H. Yang and A. Milani, On the diffusion phenomenon of quasilinear hyperbolic waves, Bull. Sci. Math., 124 (2000), 415-433. doi: 10.1016/S0007-4497(00)00141-X. Google Scholar
[37]	Qi 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. Google Scholar

[1]	Karen Yagdjian, Anahit Galstian. Fundamental solutions for wave equation in Robertson-Walker model of universe and $L^p-L^q$ -decay estimates. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 483-502. doi: 10.3934/dcdss.2009.2.483
[2]	Samer Dweik. $ L^{p, q} $ estimates on the transport density. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3001-3009. doi: 10.3934/cpaa.2019134
[3]	Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427
[4]	Maurizio Grasselli, Vittorino Pata. On the damped semilinear wave equation with critical exponent. Conference Publications, 2003, 2003 (Special) : 351-358. doi: 10.3934/proc.2003.2003.351
[5]	Sergey Zelik. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Communications on Pure & Applied Analysis, 2004, 3 (4) : 921-934. doi: 10.3934/cpaa.2004.3.921
[6]	Alexandre N. Carvalho, Jan W. Cholewa. Strongly damped wave equations in $W^(1,p)_0 (\Omega) \times L^p(\Omega)$. Conference Publications, 2007, 2007 (Special) : 230-239. doi: 10.3934/proc.2007.2007.230
[7]	Der-Chen Chang, Jie Xiao. $L^q$-Extensions of $L^p$-spaces by fractional diffusion equations. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1905-1920. doi: 10.3934/dcds.2015.35.1905
[8]	Mohamad Darwich. On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2377-2394. doi: 10.3934/cpaa.2014.13.2377
[9]	A. Kh. Khanmamedov. Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 119-138. doi: 10.3934/dcds.2011.31.119
[10]	Björn Birnir, Kenneth Nelson. The existence of smooth attractors of damped and driven nonlinear wave equations with critical exponent , s = 5. Conference Publications, 1998, 1998 (Special) : 100-117. doi: 10.3934/proc.1998.1998.100
[11]	Xinghong Pan, Jiang Xu. Global existence and optimal decay estimates of the compressible viscoelastic flows in $ L^p $ critical spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2021-2057. doi: 10.3934/dcds.2019085
[12]	Zhaojuan Wang, Shengfan Zhou. Random attractor for stochastic non-autonomous damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 545-573. doi: 10.3934/dcds.2017022
[13]	Fengjuan Meng, Chengkui Zhong. Multiple equilibrium points in global attractor for the weakly damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 217-230. doi: 10.3934/dcdsb.2014.19.217
[14]	Shengfan Zhou, Linshan Wang. Kernel sections for damped non-autonomous wave equations with critical exponent. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 399-412. doi: 10.3934/dcds.2003.9.399
[15]	Shouming Zhou. The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4967-4986. doi: 10.3934/dcds.2014.34.4967
[16]	Damiano Foschi. Some remarks on the $L^p-L^q$ boundedness of trigonometric sums and oscillatory integrals. Communications on Pure & Applied Analysis, 2005, 4 (3) : 569-588. doi: 10.3934/cpaa.2005.4.569
[17]	José Caicedo, Alfonso Castro, Rodrigo Duque, Arturo Sanjuán. Existence of $L^p$-solutions for a semilinear wave equation with non-monotone nonlinearity. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1193-1202. doi: 10.3934/dcdss.2014.7.1193
[18]	Igor Chueshov, Irena Lasiecka, Daniel Toundykov. Long-term dynamics of semilinear wave equation with nonlinear localized interior damping and a source term of critical exponent. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 459-509. doi: 10.3934/dcds.2008.20.459
[19]	Seung-Yeal Ha, Mitsuru Yamazaki. $L^p$-stability estimates for the spatially inhomogeneous discrete velocity Boltzmann model. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 353-364. doi: 10.3934/dcdsb.2009.11.353
[20]	N. V. Krylov. Some $L_{p}$-estimates for elliptic and parabolic operators with measurable coefficients. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2073-2090. doi: 10.3934/dcdsb.2012.17.2073

American Institute of Mathematical Sciences

$ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

$ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors