MR2589664 Ryo IKEHATA, Grozdena TODOROVA and Borislav YORDANOV Ryo IKEHATA, Grozdena TODOROVA and Borislav YORDANOV Critical Exponent for Semilinear Wave Equations with Space-Dependent Potential Funkcialaj Ekvacioj. Serio Internacia 52 2009 411--435 http://fe.math.kobe-u.ac.jp/FE/FullPapers/52-3/52_411.pdf http://www.ams.org/mathscinet-getitem?mr=MR2589664 We study the balance between the effect of spatial inhomogeneity of the potential in the dissipative term and the focusing nonlinearity. Sharp critical exponent results will be presented in the case of slow decaying potential. Damped wave equation, Subcritical potential, Energy decay, $L^2$-decay, Small data, Global existence and blow-up, Finite propagation speed. Primary 35L70; Secondary 35L05, 35B33, 35B40. 52-411 2009 Critical Exponent for Semilinear Wave Equations with Space-Dependent Potential Ryo IKEHATA, Grozdena TODOROVA and Borislav YORDANOV Ryo IKEHATA, Grozdena TODOROVA and Borislav YORDANOV
1 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
2 Georgiev, V.; Lindblad, H.; Sogge, C. Weighted Strichartz estimates and global existence for semilinear wave equation Amer. J. Math. 119 1997 1291--1319 MR1481816
3 Glassey, R. Existence in the large for $\Box u=F(u)$ in two space dimensions Math. Z. 178 1981 233--261 MR0631631
4 Han, Y.; Milani, A. On the diffusion phenomenon of quasilinear hyperbolic waves Bull. Sci. Math. 124 2000 415--433 MR1781556
5 Hosono, T.; Ogawa, T. Large time behavior and $L^p$-$L^q$ estimates of solutions of $2$-dimensional nonlinear damped wave equations J. Differential Equations 203 2004 82--118 MR2070387
6 Ikawa, M. Hyperbolic partial differential equations and wave phenomena Translations of Mathematical Monographs, 189, American Mathematical Society, Providence, RI 2000 MR1756774
7 Ikehata, R. Diffusion phenomenon for linear dissipative wave equations in an exterior domains J. Differential Equations 186 2002 633--651 MR1942225
8 Ikehata, R.; Miyaoka, M.; Nakatake, T. Decay estimates of solutions for dissipative wave equations in ${\bf R}^N$ with lower power nonlinearities J. Math. Soc. Japan 56 2004 365--373 MR2048464
9 Ikehata, R.; Nishihara, K. Diffusion phenomenon for second order linear evolution equations Studia Math. 158 2003 153--161 MR2013737
10 Ikehata, R.; Nishihara, K.; Zhao, H. Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption J. Differential Equations 226 2006 1--29 MR2232427
11 Ikehata, R.; Ohta, M. Critical exponent for semilinear dissipative wave equations in ${\bf R}^N$ J. Math. Anal. Appl. 269 2002 87--97 MR1907875
12 Ikehata, R.; Tanizawa, K. Global existence of solutions for semilinear damped wave equations in ${\bf R}^N$ with noncompactly supported initial data Nonlinear Anal. 61 2005 1189--1208 MR2131649
13 John, F. Blow-up of solutions of nonlinear wave equations in three space dimensions Manuscripta Math. 28 1979 235--268 MR0535704
14 Karch, G. Selfsimilar profiles in large time asymptotics of solutions to damped wave equations Studia Math. 143 2000 175--197 MR1813366
15 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
16 Deng, K.; Levine, H. A. The role of critical exponents in blow-up theorems: the sequel J. Math. Anal. Appl. 243 2000 85--126 MR1742850
17 Levine, H. A. The role of critical exponents in blowup theorems SIAM Rev. 32 1990 262--288 MR1056055
18 Li, T. T.; Zhou, Y. Breakdown of solutions to $\Box u+u_t=u^{1+\alpha}$ Discrete Contin. Dynam. Systems 1 1995 503--520 MR1357291
19 Lindblad, H.; Sogge, C., On the small-power semilinear wave equations, Preprint (1995)
20 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
21 Matsumura, A. On the asymptotic behavior of solutions of semi-linear wave equations Publ. Res. Inst. Math. Sci. 12 1976 169--189 MR0420031
22 Mochizuki, K. Scattering theory for wave equations with dissipative terms Publ. Res. Inst. Math. Sci. 12 1976/77 383--390 MR0437955
23 Mochizuki, K.; Nakazawa, H. Energy decay and asymptotic behavior of solutions to the wave equations with linear dissipation Publ. Res. Inst. Math. Sci. 32 1996 401--414 MR1409795
24 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
25 Nishihara, K. Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping J. Differential Equations 137 1997 384--395 MR1456603
26 Nishihara, K. $L^p$-$L^q$ estimates to the damped wave equation in $3$-dimensional space and their application Math. Z. 244 2003 631--649 MR1992029
27 Nishihara, K.; Zhao, H. Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption J. Math. Anal. Appl. 313 2006 598--610 MR2183322
28 Rammaha, M. Nonlinear wave equations in high dimensions Differential equations and applications, Vol. I, II (Columbus, OH, 1988), Ohio Univ. Press, Athens, OH 1989 322--326 MR1026235
29 Sideris, T., Ph.D. Thesis, Bloomington: Indiana University (1981)
30 Sideris, T. Nonexistence of global solutions to semilinear wave equations in high dimensions J. Differential Equations 52 1984 378--406 MR0744303
31 Strauss, W. Nonlinear scattering theory at low energy J. Funct. Anal. 41 1981 110--133 MR0614228
32 Strauss, W. Nonlinear wave equations C.B.M.S. Lecture Notes, 73, Amer. Math. Soc., Providence, RI 1989 MR1032250 33 Tataru, D., Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Preprint (1998) MR1804518
34 Todorova, G.; Yordanov, B. Critical exponent for a nonlinear wave equation with damping C. R. Acad. Sci. Paris Sér. I Math. 330 2000 557--562 MR1760438
35 Todorova, G.; Yordanov, B. Critical exponent for a nonlinear wave equation with damping J. Differential Equations 174 2001 464--489 MR1846744
36 Todorova, G.; Yordanov, B. Weighted $L^2$-estimates for dissipative wave equations with variable coefficients J. Differential Equations 246 2009 4497--4518 MR2523291
37 Todorova, G.; Yordanov, B. Nonlinear dissipative wave equations with potential AMS Contemp. Math., 426, Amer. Math. Soc., Providence, RI 2007 317--337 MR2311533
38 Weissler, F. Existence and non-existence of global solutions for a semilinear heat equation Israel J. Math. 38 1981 29--40 MR0599472
39 Zhang, Qi S. Nonlinear parabolic problems on manifolds, and a nonexistence result for the noncompact Yamabe problem Electron. Res. Announc. Amer. Math. Soc. 3 1997 45--51 (electronic) MR1447067
40 Zhang, Qi S. Blow-up results for nonlinear parabolic equations on manifolds Duke Math. J. 97 1999 515--539 MR1682987
41 Zhang, Qi S. 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 MR1847355
42 Zhou, Y. Cauchy problem for semilinear wave equations in four space dimensions with small initial data J. Partial Differential Equations 8 1995 135--144 MR1331521