Asymptotic Behavior for a Class of Logarithmic Wave Equations with Linear Damping
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We consider the initial boundary value problem for a class of logarithmic wave equations with linear damping. By constructing a potential well and using the logarithmic Sobolev inequality, we prove that, if the solution lies in the unstable set or the initial energy is negative, the solution will grow as an exponential function in the \(H^1_0(\Omega )\) norm as time goes to infinity. If the solution lies in a smaller set compared with the stable set, we can estimate the decay rate of the energy. These results are extensions of earlier results.
KeywordsLogarithmic wave equation Initial boundary value problem Exponential growth Energy decay
Mathematics Subject Classification35L20 35L70 35B40 35Q40
This work is supported by the National Natural Science Foundation of China (nos. 11520677, 11601122) and the Basic Research Foundation of Henan University of Technology (171164).
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