On certain classes of mild solutions of the scalar Liénard equation revisited
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914935783555072 |
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| author | Xuan, Pham Truong Van, Nguyen Thi Loan, Nguyen Thi Nguyet, Tran Minh |
| author_facet | Xuan, Pham Truong Van, Nguyen Thi Loan, Nguyen Thi Nguyet, Tran Minh |
| contents | In this work we revisit the existence, uniqueness and exponential decay of some classes of mild solutions which are almost periodic (AP-), asymptotically almost periodic (AAP-) and pseudo almost periodic (PAP-) of the scalar Linéard equation by employing the notion of Green function and Massera-type principle. First, by changing variable we convert this equation to a system of first order differential equations. Then, we transform the problem into a framework of an abstract parabolic evolution equation which associates with an evolution family equipped an exponential dichtonomy and the corresponding Green function is exponentially almost periodic. After that, we prove a Massera-type principle that the corresponding linear equation has a uniqueness AP-, AAP- and PAP- mild solution if the right hand side and the coefficient functions are AP-, AAP- and PAP- functions, respectively. The well-posedness of semilinear equation is proved by using fixed point arguments and the exponential decay of mild solutions are obtained by using Gronwall's inequality. Although our work revisits some previous works on well-posedness of the Linéard equation on these type solutions but provide a difference view by using Green function and go farther on the aspects of asymptotic behaviour of solutions and the construction of abstract theory. Our abstract results can be also applied to other parabolic evolution equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_12094 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On certain classes of mild solutions of the scalar Liénard equation revisited Xuan, Pham Truong Van, Nguyen Thi Loan, Nguyen Thi Nguyet, Tran Minh Analysis of PDEs Mathematical Physics Functional Analysis In this work we revisit the existence, uniqueness and exponential decay of some classes of mild solutions which are almost periodic (AP-), asymptotically almost periodic (AAP-) and pseudo almost periodic (PAP-) of the scalar Linéard equation by employing the notion of Green function and Massera-type principle. First, by changing variable we convert this equation to a system of first order differential equations. Then, we transform the problem into a framework of an abstract parabolic evolution equation which associates with an evolution family equipped an exponential dichtonomy and the corresponding Green function is exponentially almost periodic. After that, we prove a Massera-type principle that the corresponding linear equation has a uniqueness AP-, AAP- and PAP- mild solution if the right hand side and the coefficient functions are AP-, AAP- and PAP- functions, respectively. The well-posedness of semilinear equation is proved by using fixed point arguments and the exponential decay of mild solutions are obtained by using Gronwall's inequality. Although our work revisits some previous works on well-posedness of the Linéard equation on these type solutions but provide a difference view by using Green function and go farther on the aspects of asymptotic behaviour of solutions and the construction of abstract theory. Our abstract results can be also applied to other parabolic evolution equations. |
| title | On certain classes of mild solutions of the scalar Liénard equation revisited |
| topic | Analysis of PDEs Mathematical Physics Functional Analysis |
| url | https://arxiv.org/abs/2408.12094 |