{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,17]],"date-time":"2025-10-17T14:20:39Z","timestamp":1760710839781,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2022,1,20]],"date-time":"2022-01-20T00:00:00Z","timestamp":1642636800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The Cauchy problems of scale-invariant damped wave equations with derivative nonlinear terms and with combined nonlinear terms are studied. A new method is provided to show that the solutions will blow up in a finite time, if the nonlinear powers satisfy some conditions. The method is based on constructing appropriate test functions, by using the solution of an ordinary differential equation. It may be useful to prove the nonexistence for global solutions for other nonlinear evolution equations.<\/jats:p>","DOI":"10.3390\/sym14020198","type":"journal-article","created":{"date-parts":[[2022,1,20]],"date-time":"2022-01-20T22:59:57Z","timestamp":1642719597000},"page":"198","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A New Method for Blow-Up to Scale-Invariant Damped Wave Equations with Derivatives and Combined Nonlinear Terms"],"prefix":"10.3390","volume":"14","author":[{"given":"Yuanming","family":"Chen","sequence":"first","affiliation":[{"name":"The School of Economics, Shanghai University of Finance and Economics, Shanghai 200433, China"},{"name":"Department of Mathematics, Lishui University, Zhejiang 323000, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,20]]},"reference":[{"key":"ref_1","first-page":"321","article-title":"Symmetry analysis and exact solutions of the damped wave equation on the surface of the sphere","volume":"17","author":"Usamh","year":"2017","journal-title":"Adv. Differ. Equ. Control Process."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1032","DOI":"10.1002\/mma.3126","article-title":"The threshold of effective damping for semilinear wave equations","volume":"38","year":"2015","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"531","DOI":"10.1016\/j.jde.2021.03.033","article-title":"Small data solutions for the Euler-Poisson-Darboux equation with a power nonlinearity","volume":"286","year":"2021","journal-title":"J. Differ. Equ."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"867","DOI":"10.1515\/ans-2013-0407","article-title":"A modified test function method for damped wave equations","volume":"13","author":"Lucente","year":"2013","journal-title":"Adv. Nonlinear Stud."},{"key":"ref_5","unstructured":"D\u2019Abbicco, M., and Lucente, S. (2014, January 7\u201311). NLWE with a special scale invariant damping in odd space dimension. Proceedings of the 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications AIMS Proceedings, Madrid, Spain."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"5040","DOI":"10.1016\/j.jde.2015.06.018","article-title":"A shift in the Strauss exponent for semilinear wave equations with a not effective damping","volume":"259","author":"Lucente","year":"2015","journal-title":"J. Differ. Equ."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.1007\/s00208-018-1664-1","article-title":"Life-span of solutions to semilinear wave equation with time-dependent critical damping for specially localized initial data","volume":"372","author":"Ikeda","year":"2018","journal-title":"Math. Ann."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"269","DOI":"10.2969\/aspm\/08510269","article-title":"Weighted L2 \u2212 L2 estimate for wave equation and its applications","volume":"85","author":"Lai","year":"2020","journal-title":"Adv. Stud. Pure Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"11575","DOI":"10.1016\/j.jde.2020.08.020","article-title":"Heat-like and wave-like lifespan estimates for solutions of semilinear damped wave equations via a Kato\u2019s type lemma","volume":"269","author":"Lai","year":"2020","journal-title":"J. Differ. Equ."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1016\/j.na.2018.12.013","article-title":"Global existence and blow-up for semilinear damped wave equations in three space dimensions","volume":"182","author":"Kato","year":"2019","journal-title":"Nonlinear Anal."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"5377","DOI":"10.1016\/j.jde.2017.06.017","article-title":"Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent","volume":"263","author":"Lai","year":"2017","journal-title":"J. Differ. Equ."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"112392","DOI":"10.1016\/j.na.2021.112392","article-title":"Global existence for semilinear wave equations with scaling invariant damping in 3-D","volume":"210","author":"Lai","year":"2021","journal-title":"Nonlinear Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2680","DOI":"10.1002\/mma.5542","article-title":"A global existence result for a semilinear scale-invariant wave equation in even dimension","volume":"42","author":"Palmieri","year":"2019","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1265","DOI":"10.3934\/cpaa.2016.15.1265","article-title":"The lifespan of solutions to semilinear damped wave equations in one space dimension","volume":"15","author":"Wakasa","year":"2016","journal-title":"Commun. Pure Appl. Anal."},{"key":"ref_15","first-page":"37","article-title":"Nonexistence of global solutions of nonlinear wave equations with weak time dependent damping related to Glasseys conjecture","volume":"32","author":"Lai","year":"2019","journal-title":"Differ. Integral Equ."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"72","DOI":"10.1007\/s00526-021-01948-0","article-title":"A blow-up result for a semilinear wave equation with scale-invariant damping and mass and nonlinearity of derivative type","volume":"60","author":"Palmieri","year":"2021","journal-title":"Calc. Var. Partial. Differ. Equ."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"103275","DOI":"10.1016\/j.nonrwa.2020.103275","article-title":"Improvement on the blow-up of the wave equation with the scale-invariant damping and combined nonlinearities","volume":"59","author":"Hamouda","year":"2021","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1127","DOI":"10.1002\/mma.6817","article-title":"Blow-up for wave equation with the scale-invariant damping and combined nonlinearities","volume":"44","author":"Hamouda","year":"2021","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_19","first-page":"3","article-title":"A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities","volume":"234","author":"Mitidieri","year":"2001","journal-title":"Tr. Mat. Instituta Im. VA Steklova"},{"key":"ref_20","unstructured":"Samarsky, A.A., Galaktionov, V.A., and Kurdyumov, S.P. (1987). Regimes with Peaking in Problems for Quasilinear Parabolic Equations, Nauka."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1217","DOI":"10.1134\/S0012266114090080","article-title":"Existence and blow-up of Kantorovich principal continuous solutions of nonlinear integral equations","volume":"50","author":"Sidorov","year":"2014","journal-title":"Differ. Equ."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"5165","DOI":"10.1016\/j.jde.2019.05.029","article-title":"Blow-up phenomena of semilinear wave equations and their weakly coupled systems","volume":"267","author":"Ikeda","year":"2019","journal-title":"J. Differ. Equ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"124189","DOI":"10.1016\/j.jmaa.2020.124189","article-title":"Strauss exponent for semilinear wave equations with scattering space dependent damping","volume":"489","author":"Lai","year":"2020","journal-title":"J. Math. Anal. Appl."},{"key":"ref_24","unstructured":"Lai, N., and Schiavone, M. (2007). Lifespan estimate for semilinear generalized Tricomi equations. arXiv."},{"key":"ref_25","unstructured":"Abramowitz, M., and Stegun, I.A. (1965). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Table, Dover."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/2\/198\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:04:25Z","timestamp":1760133865000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/14\/2\/198"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,20]]},"references-count":25,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2022,2]]}},"alternative-id":["sym14020198"],"URL":"https:\/\/doi.org\/10.3390\/sym14020198","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2022,1,20]]}}}