Previous |  Up |  Next


Singular ordinary differential equation of the second order; lower and upper functions; time singularities; unbounded domain; homoclinic solution
The paper investigates singular nonlinear problems arising in hydrodynamics. In particular, it deals with the problem on the half-line of the form \[(p(t)u^{\prime }(t))^{\prime } = p(t)f(u(t)),\] \[u^{\prime }(0) = 0,\quad u(\infty ) = L.\] The existence of a strictly increasing solution (a homoclinic solution) of this problem is proved by the dynamical systems approach and the lower and upper functions method.
[1] Berestycki, H., Lions, P. L., Peletier, L. A.: An ODE approach to the existence of positive solutions for semilinear problems in $\cal R^N$. Indiana University Mathematics Journal 30, 1 (1981), 141–157. MR 0600039
[2] Bonheure, D., Gomes, J. M., Sanchez, L.: Positive solutions of a second–order singular ordinary differential equation. Nonlinear Analysis 61 (2005), 1383–1399. MR 2135816 | Zbl 1109.34310
[3] Dell’Isola, F., Gouin, H., Rotoli, G.: Nucleation of spherical shell-like interfaces by second gradient theory: numerical simulations. Eur. J. Mech B/Fluids 15 (1996), 545–568.
[4] Gouin, H., Rotoli, G.: An analytical approximation of density profile and surface tension of microscopic bubbles for Van der Waals fluids. Mech. Research Communic. 24 (1997), 255–260. Zbl 0899.76064
[5] Kitzhofer, G., Koch, O., Lima, P., Weinmüller, E.: Efficient numerical solution of the density profile equation in hydrodynamics. J. Sci. Comput. 32, 3 (2007), 411–424. MR 2335787 | Zbl 1179.76062
[6] Koch, O., Kofler, P., Weinmüller, E.: Initial value problems for systems of ordinary first and second order differential equations with a singularity of the first kind. Analysis 21 (2001), 373–389. MR 1867622 | Zbl 1029.34002
[7] Lima, P. M., Chemetov, N. V., Konyukhova, N. B., Sukov, A. I.: Analytical–numerical investigation of bubble-type solutions of nonlinear singular problems. J. Comp. Appl. Math. 189 (2006), 260–273. MR 2202978 | Zbl 1100.65066
[8] Rachůnková, I., Koch, O., Pulverer, G., Weinmüller, E.: On a singular boundary value problem arising in the theory of shallow membrane caps. J. Math. Anal. Appl. 332 (2007), 532–541. MR 2319681 | Zbl 1118.34013
Partner of
EuDML logo