{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,15]],"date-time":"2026-02-15T08:56:38Z","timestamp":1771145798379,"version":"3.50.1"},"reference-count":31,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2023,1,1]],"date-time":"2023-01-01T00:00:00Z","timestamp":1672531200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11801056"],"award-info":[{"award-number":["11801056"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"Natural Science Foundation of China","doi-asserted-by":"publisher","award":["JDL2020027"],"award-info":[{"award-number":["JDL2020027"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Research Project of the Educational Department of Liaoning Province","award":["11801056"],"award-info":[{"award-number":["11801056"]}]},{"name":"Natural Science Research Project of the Educational Department of Liaoning Province","award":["JDL2020027"],"award-info":[{"award-number":["JDL2020027"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Considered herein is the initial-boundary value problem for a semilinear parabolic equation with a memory term and non-local source wt\u2212\u0394Bw\u2212\u0394Bwt+\u222b0tg(t\u2212\u03c4)\u0394Bw(\u03c4)d\u03c4=|w|p\u22121w\u22121|B|\u222bB|w|p\u22121wdx1x1dx\u2032 on a manifold with conical singularity, where the Fuchsian type Laplace operator \u0394B is an asymmetry elliptic operator with conical degeneration on the boundary x1=0. Firstly, we discuss the symmetrical structure of invariant sets with the help of potential well theory. Then, the problem can be decomposed into two symmetric cases: if w0\u2208W and \u03a0(w0)&gt;0, the global existence for the weak solutions will be discussed by a series of energy estimates under some appropriate assumptions on the relaxation function, initial data and the symmetric structure of invariant sets. On the contrary, if w0\u2208V and \u03a0(w0)&lt;0, the nonexistence of global solutions, i.e., the solutions blow up in finite time, is obtained by using the convexity technique.<\/jats:p>","DOI":"10.3390\/sym15010122","type":"journal-article","created":{"date-parts":[[2023,1,2]],"date-time":"2023-01-02T02:12:48Z","timestamp":1672625568000},"page":"122","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Nonlocal Pseudo-Parabolic Equation with Memory Term and Conical Singularity: Global Existence and Blowup"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9233-2798","authenticated-orcid":false,"given":"Jiali","family":"Yu","sequence":"first","affiliation":[{"name":"School of Science, Dalian Jiaotong University, Dalian 116028, China"}]},{"given":"Jihong","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Science, Dalian Jiaotong University, Dalian 116028, China"}]}],"member":"1968","published-online":{"date-parts":[[2023,1,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"718","DOI":"10.1137\/0153036","article-title":"Blow-up in a partial differential equation with conserved first integral","volume":"53","author":"Budd","year":"1993","journal-title":"SIAM J. Appl. Math."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF02844682","article-title":"Semilinear parabolic equations with prescribed energy","volume":"44","author":"Hu","year":"1995","journal-title":"Rend. Circ. Mat. Palermo"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.anihpc.2005.09.005","article-title":"A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions","volume":"24","author":"Jazar","year":"2007","journal-title":"Ann. Inst. H Poincar\u00e9 Anal. Non Lin\u00e9aire"},{"key":"ref_4","first-page":"215","article-title":"Blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions","volume":"525","author":"Jazar","year":"2008","journal-title":"Ann. Inst. H Poincar\u00e9 Anal. Non Lin\u00e9aire"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/0022-0396(88)90116-7","article-title":"Total blow-up versus single point blow-up","volume":"73","author":"Bebernes","year":"1988","journal-title":"J. Differ. Equ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/BF00276081","article-title":"Non-local interactions in population dynamics","volume":"27","author":"Furter","year":"1989","journal-title":"J. Math. Biol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1155\/2013\/643819","article-title":"Blow-up in a slow diffusive p-Laplace equation with the Neumann boundary conditions","volume":"2013","author":"Qu","year":"2013","journal-title":"Abstr. Appl. Anal."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1016\/j.jmaa.2013.10.040","article-title":"Blow-up and extinction in a nonlocal p-Laplace equation with Neumann boundary conditions","volume":"412","author":"Qu","year":"2014","journal-title":"J. Math. Anal. Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1527","DOI":"10.1016\/j.jmaa.2014.09.006","article-title":"Non-extinction of solutions to a fast diffusive p-Laplace equation with Neumann boundary conditions","volume":"422","author":"Guo","year":"2015","journal-title":"J. Math. Anal. Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"896","DOI":"10.1016\/j.camwa.2015.06.003","article-title":"Blow-up phenomena for a nonlocal semilinear parabolic equation with positive initial energy","volume":"70","author":"Khelghati","year":"2015","journal-title":"Comput. Math. Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.aml.2016.05.002","article-title":"Global existence and non-extinction of solutions to a fourth-order parabolic equation","volume":"61","author":"Cao","year":"2016","journal-title":"Appl. Math. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.na.2016.08.021","article-title":"Global existence blow up and extinction for a class of thin-film equation","volume":"147","author":"Li","year":"2016","journal-title":"Nonlinear Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1133","DOI":"10.1016\/j.jmaa.2016.09.026","article-title":"Blow-up for a thin-film equation with positive initial energy","volume":"446","author":"Zhou","year":"2017","journal-title":"J. Math. Anal. Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"198","DOI":"10.1016\/j.aml.2016.09.007","article-title":"Blowup, extinction and non-extinction for a nonlocal p-biharmonic parabolic equation","volume":"64","author":"Hao","year":"2017","journal-title":"Appl. Math. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.jmaa.2017.08.047","article-title":"Finite time blow-up for a thin-film equation with initial data at arbitrary energy level","volume":"458","author":"Sun","year":"2018","journal-title":"J. Math. Anal. Appl."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1016\/j.jmaa.2017.09.031","article-title":"Global existence and finite time blow-up of the solution for a thin-film equation with high initial energy","volume":"458","author":"Xu","year":"2018","journal-title":"J. Math. Anal. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1007\/s10455-010-9226-0","article-title":"Existence theorem for a class of semilinear totally characteristic elliptic equations with critical cone Sobolev exponents","volume":"39","author":"Chen","year":"2011","journal-title":"Ann. Glob. Anal. Geom."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1007\/s00526-011-0418-7","article-title":"Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on a manifold with conical singularities","volume":"43","author":"Chen","year":"2012","journal-title":"Calc. Var. Partial Differ. Equ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"329","DOI":"10.1007\/s11868-012-0046-9","article-title":"Global existence and nonexistence for semilinear parabolic equations with conical degeneration","volume":"3","author":"Chen","year":"2012","journal-title":"J. Pseudo-Differ. Oper. Appl."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"629","DOI":"10.1007\/s11868-017-0216-x","article-title":"Global existence, exponential decay and finite time blow-up of solutions for a class of semilinear pseudo-parabolic equations with conical degeneration","volume":"8","author":"Li","year":"2017","journal-title":"J. Pseudo. Differ. Oper. Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4566","DOI":"10.1016\/j.jde.2020.03.030","article-title":"Global well-posedness for a nonlocal semilinear pseudo-parabolic equation with conical degeneration","volume":"269","author":"Di","year":"2020","journal-title":"J. Differ. Equ."},{"key":"ref_22","first-page":"22","article-title":"Multiple positive solutions for degenerate elliptic equations with critical cone Sobolev exponents on singular manifolds","volume":"181","author":"Fan","year":"2013","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2331","DOI":"10.1016\/j.jfa.2013.07.013","article-title":"Existence result for a class of semilinear totally characteristic hypoelliptic equations with conical degeneration","volume":"265","author":"Alimohammady","year":"2013","journal-title":"J. Funct. Anal."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1016\/j.jmaa.2017.05.057","article-title":"Invariance and existence analysis for semilinear hyperbolic equations with damping and conical singularity","volume":"455","author":"Alimohammady","year":"2017","journal-title":"J. Math. Anal. Appl."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"4160","DOI":"10.1002\/mma.4295","article-title":"Global results for semilinear hyperbolic equations with damping term on manifolds with conical singularity","volume":"40","author":"Alimohammady","year":"2017","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1007\/s13226-017-0215-x","article-title":"Multiple solutions for non-homogenous degenerate Sch\u00f6rdinger equations in cone Sobolev spaces","volume":"48","author":"Alimohammady","year":"2017","journal-title":"Indian J. Pure Appl. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1007\/s00526-022-02316-2","article-title":"Global well-posedness for a class of semilinear hyperbolic equations with singular potentials onmanifolds with conical singularities","volume":"61","author":"Luo","year":"2022","journal-title":"Calc. Var."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Alshin, A.B., Korpusov, M.O., and Siveshnikov, A.G. (2011). Blow Up in Nonlinear Sobolev Type Equations, Walter de Gruyter.","DOI":"10.1515\/9783110255294"},{"key":"ref_29","first-page":"1031","article-title":"Generalization of equations of motion of underground water with free surface","volume":"202","author":"Dzektser","year":"1972","journal-title":"Dokl. Akad. Nauk SSSR"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1007\/s13324-020-00442-8","article-title":"Well-posedness for Hardy-H\u00e9non parabolic equations with fractional Brownian noise","volume":"11","author":"Majdoub","year":"2021","journal-title":"Anal. Math. Phys."},{"key":"ref_31","first-page":"12","article-title":"Existence and stability for fractional parabolic integro-partial differential equations with fractional Brownian motion and nonlocal condition","volume":"5","author":"Ahmed","year":"2018","journal-title":"Cogent Math. Stat."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/122\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:55:14Z","timestamp":1760118914000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/1\/122"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,1]]},"references-count":31,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,1]]}},"alternative-id":["sym15010122"],"URL":"https:\/\/doi.org\/10.3390\/sym15010122","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,1]]}}}