Cauchy problem for quasi-linear wave equations with nonlinear damping and source terms |
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Authors: | Zhijian Yang |
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Institution: | Department of Mathematics, Zhengzhou University, No. 75, Daxue Road, Zhengzhou 450052, PR China |
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Abstract: | The paper studies the existence and non-existence of global weak solutions to the Cauchy problem for a class of quasi-linear wave equations with nonlinear damping and source terms. It proves that when α?max{m,p}, where m+1, α+1 and p+1 are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, under rather mild conditions on initial data, the Cauchy problem admits a global weak solution. Especially in the case of space dimension N=1, the weak solutions are regularized and so generalized and classical solution both prove to be unique. On the other hand, if 0?α<1, and the initial energy is negative, then under certain opposite conditions, any weak solution of the Cauchy problem blows up in finite time. And an example is shown. |
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Keywords: | Cauchy problem Quasi-linear wave equation Global solutions Blowup of solutions |
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