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Sub-solutions and a point-wise Hopf's lemma for fractional $ p $-Laplacian
Classification and evolution of bifurcation curves for a porous-medium combustion problem with large activation energy
1. | Fundamental General Education Center, National Chin-Yi University of Technology, Taichung 411, Taiwan |
2. | Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan |
$ \begin{equation*} \begin{cases} u^{\prime \prime }(x)+\lambda \dfrac{1+au}{1+e^{d(1-u)}} = 0, \ -1<x<1, \\ u(-1) = u(1) = 0, \end{cases} \end{equation*} $ |
$ u $ |
$ \lambda >0 $ |
$ a\geq 0 $ |
$ d>0 $ |
$ (\lambda , ||u||_{\infty }) $ |
$ \ (d, a)\in \Omega \equiv \left \{ (d, a):(0<d<d_{1}, \text{ }a\geq A_{1}(d))\text{ or }(d\geq d_{1}, \text{ }a\geq 0)\right \} $ |
$ d_{1}\approx 2.225 $ |
$ A_{1}(d) $ |
$ (0, d_{1}]. $ |
$ \ (d, a)\in \Omega , $ |
$ d>0 $ |
$ a\geq \tilde{a}\approx 1.704 $ |
$ \tilde{a}, $ |
$ S $ |
$ (\lambda , \left \Vert u\right \Vert _{\infty }) $ |
References:
[1] |
A. Friedman and A. E. Tzavaras, Combustion in a porous medium, SIAM J. Math. Anal., 19 (1988), 509–519.
doi: 10.1137/0519036. |
[2] |
K. C. Hung and S. H. Wang, A theorem on S-shaped bifurcation curve for a positone problem with convex-concave nonlinearity and its applications to the perturbed Gelfand problem, J. Differ. Equ., 251 (2011), 223–237.
doi: 10.1016/j.jde.2011.03.017. |
[3] |
T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1–13.
doi: 10.1512/iumj.1970.20.20001. |
[4] |
P. Nistri, Positive solutions of a nonlinear eigenvalue problem with discontinuous nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 83 (1979), 133–145.
doi: 10.1017/S0308210500011458. |
[5] |
J. Norbury and A. M. Stuart, Parabolic free boundary problems arising in porous medium combustion, IMA J. Appl. Math., 39 (1987), 241–257.
doi: 10.1093/imamat/39.3.241. |
[6] |
J. Norbury and A. M. Stuart, A model for porous-medium combustion, Quart. J. Mech. Appl. Math., 42 (1989), 159–178.
doi: 10.1093/qjmam/42.1.159. |
[7] |
K. Scott, The Smouldering of Peat, Ph.D. Dissertation, University of Manchester, England, (2013), 178 pp. Google Scholar |
[8] |
S. H. Wang, Bifurcation of an equation arising in porous-medium combustion, IMA J. Appl. Math., 56 (1996), 219–234.
doi: 10.1093/imamat/56.3.219. |
[9] |
S. H. Wang and T. S. Yeh, A complete classification of bifurcation diagrams of a Dirichlet problem with concave-convex nonlinearities, J. Math. Anal. Appl., 291 (2004), 128–153.
doi: 10.1016/j.jmaa.2003.10.021. |
show all references
References:
[1] |
A. Friedman and A. E. Tzavaras, Combustion in a porous medium, SIAM J. Math. Anal., 19 (1988), 509–519.
doi: 10.1137/0519036. |
[2] |
K. C. Hung and S. H. Wang, A theorem on S-shaped bifurcation curve for a positone problem with convex-concave nonlinearity and its applications to the perturbed Gelfand problem, J. Differ. Equ., 251 (2011), 223–237.
doi: 10.1016/j.jde.2011.03.017. |
[3] |
T. Laetsch, The number of solutions of a nonlinear two point boundary value problem, Indiana Univ. Math. J., 20 (1970), 1–13.
doi: 10.1512/iumj.1970.20.20001. |
[4] |
P. Nistri, Positive solutions of a nonlinear eigenvalue problem with discontinuous nonlinearity, Proc. Roy. Soc. Edinburgh Sect. A, 83 (1979), 133–145.
doi: 10.1017/S0308210500011458. |
[5] |
J. Norbury and A. M. Stuart, Parabolic free boundary problems arising in porous medium combustion, IMA J. Appl. Math., 39 (1987), 241–257.
doi: 10.1093/imamat/39.3.241. |
[6] |
J. Norbury and A. M. Stuart, A model for porous-medium combustion, Quart. J. Mech. Appl. Math., 42 (1989), 159–178.
doi: 10.1093/qjmam/42.1.159. |
[7] |
K. Scott, The Smouldering of Peat, Ph.D. Dissertation, University of Manchester, England, (2013), 178 pp. Google Scholar |
[8] |
S. H. Wang, Bifurcation of an equation arising in porous-medium combustion, IMA J. Appl. Math., 56 (1996), 219–234.
doi: 10.1093/imamat/56.3.219. |
[9] |
S. H. Wang and T. S. Yeh, A complete classification of bifurcation diagrams of a Dirichlet problem with concave-convex nonlinearities, J. Math. Anal. Appl., 291 (2004), 128–153.
doi: 10.1016/j.jmaa.2003.10.021. |




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