MR2271233 Ono, Kosuke Kosuke ONO Global Solvability and $L^p$ Decay for the Semilinear Dissipative Wave Equations in Four and Five Dimensions Funkcialaj Ekvacioj. Serio Internacia 49 2006 215--233 http://fe.math.kobe-u.ac.jp/FE/FullPapers/49-2/49_215.pdf http://www.ams.org/mathscinet-getitem?mr=MR2271233 We study the global existence, uniqueness, and asymptotic behavior of solutions to the Cauchy problem for the semilinear dissipative wave equations: $(\square+\partial_t)u=|u|^{\alpha+1}$ in $R^N\times(0,\infty)$ with $u|_{t=0}=\varepsilon u_0$ and $\partial_t u|_{t=0}=\varepsilon u_1$ for a small parameter $\varepsilon>0$. Here, we do not assume any compactly support conditions on the initial data $(u_0,u_1)$. When dimension $N=4,5$ and $\alpha$ is greater than a critical number $2/N$ which is often called Fujita's exponent, we solve the global in time solvability problem and we derive the sharp decay rates of $L^p$ norm with $p\ge1$ of the solutions. Global existence, Decay, Wave equations, Damping, Dissipation, Fujita's exponent. 35L05, 35B40, 35B45. 49-215 2006 Global Solvability and $L^p$ Decay for the Semilinear Dissipative Wave Equations in Four and Five Dimensions ONO, Kosuke Kosuke ONO 1 Cazenave, T.; Haraux, A. An Introduction to Semilinear Evolution Equations Oxford Science Publications 1998 MR1691574 2 Courant, R.; Hilbert, D. Methods of Mathematical Physics II J. Wiley & Sons, New York 1989 MR1013360 3 Evans, L. C. Partial Differential Equations Math., Amer. Math. Soc., Providence, RI 1998 MR1625845
4 Fujita, H. On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$ J. Fac. Sci. Univ. Tokyo, Sect., I 13 1966 109-124 MR0214914
5 Hayashi, N.; Kaikina, E. I.; Naumkin, P. I. Damped wave equation with supper critical Differential Integral Equations 17 2004 637-652 MR2054939
6 Hosono, T.; Ogawa, T. 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 MR2070387
7 Ikehata, R.; Miyaoka, Y.; Nakatake, T. Decay estimates of solutions for dissipative wave equations in $R^N$ with lower power nonlinearities J. Math. Soc. Japan 56 2004 365-373 MR2048464
8 Ikehata, R.; Ohta, M. Critical exponents for semilinear dissipative wave equations in $R^N$ J. Math. Anal. Appl. 269 2002 87-97 MR1907875
9 Kawashima, S.; Nakao, M.; Ono, K. 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 MR1348752
10 Kirane, M.; Qafsaoui, M. Fujita's exponent for a semilinear wave equation with linear damping Adv. Nonlinear Stud. 2 2002 41-49 MR1881998
11 Li, T.-T.; Zhou, Y. Breakdown of the solutions to $\square u+u_t=|u|^{1+\alpha}$ Discrete. Contin. Dynam. Systems 1 1995 503-520 MR1357291
12 Marcati, P.; Nishihara, K. 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 MR1978385
13 Matsumura, A. On the asymptotic behavior of solutions of semi-linear wave equations Publ. R.I.M.S., Kyoto Univ. 12 1976/77 169-189 MR0420031
14 Milani, A.; Han, Y. $L^1$ Decay estimates for dissipative wave equations Math. Methods Appl. Sci. 24 2001 319-338 MR1818898
15 Nakao, M.; Ono, K. Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations Math. Z. 214 1993 325-342 MR1240892
16 Narazaki, T. $L^p-L^q$ estimates for damped wave equations and their applications to semi-linear problem J. Math. Soc. Japan 56 2004 585-626 MR2048476
17 Nishihara, K. $L^p-L^q$ estimates of solutions to the damped wave equation in 3-dimensional space and their application Math. Z. 244 2003 631-649 MR1992029
18 Ono, K. Asymptotic behavior of solutions for semilinear telegraph equations J. Math. Tokushima Univ. 31 1997 11-22 MR1618990
19 Ono, K. Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations Discrete. Contin. Dynam. Systems 9 2003 651-662 MR1974531
20 Strauss, W. A. Nonlinear wave equations CBMS Regional Conf. Ser. in Math., Amer. Math. Soc., Providence, RI 1989 MR1032250
21 Todorova, G.; Yordanov, B. Critical exponent for a nonlinear wave equation with damping J. Differential Equations 174 2001 464-489 MR1846744
22 Zhang, Q. A blow-up result for a nonlinear wave equation with damping: the critical case C. R. Acad. Sci. Paris 333 2001 109-114 MR1847355