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Zhang, Jialing

University of British Columbia

First, we consider the initial boundary value problem for a class of pseudo-parabolic equations with combined nonlinearities. Employing the simplified potential well method, we obtain the existence of a global weak solution and a finite time blow-up phenomenon. By a two-step trick, we prove that the global weak solution has exponential decay. A conclusion we derived is that the power-type nonlinearity breaks the infinite time blow-up phenomenon caused by the logarithmic nonlinearity. In addition, we estimate an upper bound and a lower bound for the blow-up time under certain conditions. Second, we study the initial boundary value problem for a class of fourth order parabolic equations with combined logarithmic nonlinearities. Using the simplified potential well method and the logarithmic Sobolev inequality, we prove the existence and blow-up results of weak solutions. The blow-up phenomenon implies that a class of logarithmic nonlinearity is important for the solutions to blow up in finite time. Applying the concavity method, an upper bound for the blow-up time is estimated. We provide a lower bound for the blow-up time by combining the interpolation inequality with the Sobolev inequality. Finally, we consider the Dirichlet problem for a class of degenerate fourth order elliptic equations with singular potential on conical singular manifolds. We introduce the background of conical singular manifolds and the weighted cone Sobolev spaces. For the singular potential term, we establish the Rellich inequality on manifolds with conical singularities, which is used to get the equivalent norm on a suitable weighted cone Sobolev space. Applying the variational method, we obtain the existence of multiple weak solutions in a suitable weighted cone Sobolev space. Moreover, we show that the domain of the degenerate operator is a direct sum of an infinite-dimensional space and a finite-dimensional space of functions with specific asymptotics near the cone point.

Text

2025-04-17

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/90680

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/