Asymptotic behaviour of solutions to nonlinear parabolic equations with variable viscosity and geometric terms
In this paper we study the asymptotic behaviour of solutions of certain nonlinear parabolic equations with variable viscosity and geometric terms. We generalize the results on the large time behaviour and vanishing viscosity limits obtained earlier for planar Burgers equation by Hopf [7] Lighthi...
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| Format: | Article |
| Language: | English |
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Texas State University
2007-11-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2007/157/abstr.html |
| _version_ | 1811343701857796096 |
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| author | Kayyunnapara Thomas Joseph |
| author_facet | Kayyunnapara Thomas Joseph |
| author_sort | Kayyunnapara Thomas Joseph |
| collection | DOAJ |
| description | In this paper we study the asymptotic behaviour of solutions of certain nonlinear parabolic equations with variable viscosity and geometric terms. We generalize the results on the large time behaviour and vanishing viscosity limits obtained earlier for planar Burgers equation by Hopf [7] Lighthill [20] and others. For several classes of systems of equations we derive explicit solution for initial value problem with different types of initial conditions and study large time behaviour of the solutions and its asymptotic form. We derive the simple hump solutions and $N$-wave solutions as its asymptotes depending on the conditions on the data and derive $L^p$ decay estimates for solutions and show that they depend on the variable viscosity coefficient and geometric terms. We also analyse the small viscosity limit of these solutions. |
| first_indexed | 2024-04-13T19:33:28Z |
| format | Article |
| id | doaj.art-8d8976ecde60495e9462ddf67cfed616 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-04-13T19:33:28Z |
| publishDate | 2007-11-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-8d8976ecde60495e9462ddf67cfed6162022-12-22T02:33:06ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912007-11-012007157123Asymptotic behaviour of solutions to nonlinear parabolic equations with variable viscosity and geometric termsKayyunnapara Thomas JosephIn this paper we study the asymptotic behaviour of solutions of certain nonlinear parabolic equations with variable viscosity and geometric terms. We generalize the results on the large time behaviour and vanishing viscosity limits obtained earlier for planar Burgers equation by Hopf [7] Lighthill [20] and others. For several classes of systems of equations we derive explicit solution for initial value problem with different types of initial conditions and study large time behaviour of the solutions and its asymptotic form. We derive the simple hump solutions and $N$-wave solutions as its asymptotes depending on the conditions on the data and derive $L^p$ decay estimates for solutions and show that they depend on the variable viscosity coefficient and geometric terms. We also analyse the small viscosity limit of these solutions.http://ejde.math.txstate.edu/Volumes/2007/157/abstr.htmlParabolic equationsexact solutionsasymptotic behaviour |
| spellingShingle | Kayyunnapara Thomas Joseph Asymptotic behaviour of solutions to nonlinear parabolic equations with variable viscosity and geometric terms Electronic Journal of Differential Equations Parabolic equations exact solutions asymptotic behaviour |
| title | Asymptotic behaviour of solutions to nonlinear parabolic equations with variable viscosity and geometric terms |
| title_full | Asymptotic behaviour of solutions to nonlinear parabolic equations with variable viscosity and geometric terms |
| title_fullStr | Asymptotic behaviour of solutions to nonlinear parabolic equations with variable viscosity and geometric terms |
| title_full_unstemmed | Asymptotic behaviour of solutions to nonlinear parabolic equations with variable viscosity and geometric terms |
| title_short | Asymptotic behaviour of solutions to nonlinear parabolic equations with variable viscosity and geometric terms |
| title_sort | asymptotic behaviour of solutions to nonlinear parabolic equations with variable viscosity and geometric terms |
| topic | Parabolic equations exact solutions asymptotic behaviour |
| url | http://ejde.math.txstate.edu/Volumes/2007/157/abstr.html |
| work_keys_str_mv | AT kayyunnaparathomasjoseph asymptoticbehaviourofsolutionstononlinearparabolicequationswithvariableviscosityandgeometricterms |