On a class of forward-backward parabolic equations: formation of singularities. (English) Zbl 1443.35066
Summary: We study the formation of singularities for the problem
\[
\begin{cases} u_t = [\varphi (u)]_{xx} + \varepsilon [\psi (u)]_{txx} & \text{in } \Omega \times (0, T) \\
\varphi (u) + \varepsilon [\psi (u)]_t = 0 & \text{in } \partial \Omega \times (0, T) \\
u = u_0 \geq 0 & \text{in } \Omega \times \{0\},
\end{cases}
\]
where \(\varepsilon\) and \(T\) are positive constants, \(\Omega\) a bounded interval, \(u_0\) a nonnegative Radon measure on \(\Omega, \phi\) a nonmonotone and nonnegative function with \(\varphi(0) = \varphi(\infty) = 0\), and \(\psi\) an increasing bounded function. We show that if \(u_0\) is a bounded or continuous function, singularities may appear spontaneously. The class of singularities which can arise in finite time is remarkably large, and includes infinitely many Dirac masses and singular continuous measures.
MSC:
| 35K55 | Nonlinear parabolic equations |
| 35R25 | Ill-posed problems for PDEs |
| 28A33 | Spaces of measures, convergence of measures |
| 28A50 | Integration and disintegration of measures |
Keywords:
forward-backward parabolic equations; formation of singularities; pseudo-parabolic regularization; Radon measuresReferences:
| [1] | Baras, P.; Pierre, M., Singularités éliminables pour des équations semilinéaires, Ann. Inst. Fourier, 34, 185-206 (1984) · Zbl 0519.35002 |
| [2] | Barenblatt, G. I.; Bertsch, M.; Dal Passo, R.; Prostokishin, V. M.; Ughi, M., A mathematical problem of turbulent heat and mass transfer in stably stratified turbulent shear flow, J. Fluid Mech., 253, 341-358 (1993) · Zbl 0777.76041 |
| [3] | Barenblatt, G. I.; Bertsch, M.; Dal Passo, R.; Ughi, M., A degenerate pseudoparabolic regularization of a nonlinear forward-backward heat equation arising in the theory of heat and mass exchange in stably stratified turbulent shear flow, SIAM J. Math. Anal., 24, 1414-1439 (1993) · Zbl 0790.35054 |
| [4] | Bellettini, G.; Bertini, L.; Mariani, M.; Novaga, M., Convergence of the one-dimensional Cahn-Hilliard equation, SIAM J. Math. Anal., 44, 3458-3480 (2012) · Zbl 1277.35197 |
| [5] | Bellettini, G.; Fusco, G.; Guglielmi, N., A concept of solution and numerical experiments for forward-backward diffusion equations, Discrete Contin. Dyn. Syst., 16, 783-842 (2006) · Zbl 1105.35007 |
| [6] | Bénilan, Ph.; Brezis, H., Nonlinear problems related to the Thomas-Fermi equation, J. Evol. Equ., 3, 673-770 (2004) · Zbl 1150.35406 |
| [7] | Bénilan, Ph.; Brezis, H.; Crandall, M., A semilinear equation in \(L^1( \mathbb{R}^N)\), Ann. Sc. Norm. Pisa, 2, 523-555 (1975) · Zbl 0314.35077 |
| [8] | Bertsch, M.; Giacomelli, L.; Tesei, A., Measure-valued solutions to a nonlinear fourth-order regularization of forward-backward parabolic equations, SIAM J. Math. Anal., 51, 374-402 (2019) · Zbl 1407.35104 |
| [9] | Bertsch, M.; Smarrazzo, F.; Tesei, A., Pseudo-parabolic regularization of forward-backward parabolic equations: a logarithmic nonlinearity, Anal. PDE, 6, 1719-1754 (2013) · Zbl 1284.35121 |
| [10] | Bertsch, M.; Smarrazzo, F.; Tesei, A., On a class of forward-backward parabolic equations: existence of solutions, Nonlinear Anal., 177, 45-87 (2018) · Zbl 1401.35160 |
| [11] | Bertsch, M.; Smarrazzo, F.; Tesei, A., On a class of forward-backward parabolic equations: properties of solutions, SIAM J. Math. Anal., 49, 2037-2060 (2017) · Zbl 1372.35323 |
| [12] | Bertsch, M.; Smarrazzo, F.; Tesei, A., On a pseudoparabolic regularization of a forward-backward-forward equation, Nonlinear Anal., 129, 217-257 (2015) · Zbl 1345.35058 |
| [13] | Bertsch, M.; Smarrazzo, F.; Tesei, A., Pseudo-parabolic regularization of forward-backward parabolic equations: power-type nonlinearities, J. Reine Angew. Math., 712, 51-80 (2016) · Zbl 1377.35164 |
| [14] | Bertsch, M.; Smarrazzo, F.; Tesei, A., Nonuniqueness of solutions for a class of forward-backward parabolic equations, Nonlinear Anal., 137, 190-212 (2016) · Zbl 1339.35008 |
| [15] | Binder, K.; Frisch, H. L.; Jäckle, J., Kinetics of phase separation in the presence of slowly relaxing structural variables, J. Chem. Phys., 85, 1505-1512 (1986) |
| [16] | Brezis, H., Équations et inéquations non linéaires dans les espaces vectoriels en dualité, Ann. Inst. Fourier (Grenoble), 18, 115-175 (1968) · Zbl 0169.18602 |
| [17] | Brokate, M.; Sprekels, J., Hysteresis and Phase Transitions, Applied Mathematical Sciences, vol. 121 (1996), Springer · Zbl 0951.74002 |
| [18] | Brezis, H.; Marcus, M.; Ponce, A. C., Nonlinear elliptic equations with measures revisited. Mathematical aspects of nonlinear dispersive equations, Ann. Math. Stud., 163, 55-109 (2007) · Zbl 1151.35034 |
| [19] | Cao, X.; Pop, I. S., Uniqueness of weak solutions for a pseudoparabolic equation modeling two phase flow in porous media, Appl. Math. Lett., 46, 25-30 (2015) · Zbl 1524.35333 |
| [20] | Dupaigne, L.; Ponce, A. C.; Porretta, A., Elliptic equations with vertical asymptotes in the nonlinear term, J. Anal. Math., 98, 349-396 (2006) · Zbl 1132.35366 |
| [21] | Fan, Y.; Pop, I. S., A class of degenerate pseudoparabolic equations: existence, uniqueness of weak solutions and error estimates for the Euler-implicit discretization, Math. Methods Appl. Sci., 34, 2329-2339 (2011) · Zbl 1235.35182 |
| [22] | Giaquinta, M.; Modica, G.; Souček, J., Cartesian Currents in the Calculus of Variations, vol. I (1998), Springer · Zbl 0914.49001 |
| [23] | Gurtin, M., Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92, 178-192 (1996) · Zbl 0885.35121 |
| [24] | Helmers, M.; Herrmann, M., Interface dynamics in discrete forward-backward diffusion equations, Multiscale Model. Simul., 11, 1261-1297 (2013) · Zbl 1302.35199 |
| [25] | Höllig, K., Existence of infinitely many solutions for a forward backward heat equation, Trans. Am. Math. Soc., 278, 299-316 (1983) · Zbl 0524.35057 |
| [26] | Khomrutai, S., Uniqueness and grow-up rate of solutions for pseudo-parabolic equations in \(\mathbb{R}^n\) with a sublinear source, Appl. Math. Lett., 48, 8-13 (2015) · Zbl 1388.35118 |
| [27] | Leray, J.; Lions, J.-L., Quelques rèsultats de Višik sur les problèmes elliptiques non linèaires par les mèthodes de Minty-Browder, Bull. Soc. Math. Fr., 93, 97-107 (1965) · Zbl 0132.10502 |
| [28] | Marras, M.; Vernier-Piro, S.; Viglialoro, G., Blow-up phenomena for nonlinear pseudoparabolic equations with gradient term, Discrete Contin. Dyn. Syst., Ser. B, 22, 2291-2300 (2017) · Zbl 1366.35101 |
| [29] | Mikelić, A., A global existence result for the equation describing unsaturated flow in porous media with dynamic capillary pressure, J. Differ. Equ., 248, 1561-1577 (2010) · Zbl 1191.35156 |
| [30] | Novick-Cohen, A., On the viscous Cahn-Hilliard equation, (Ball, J. M., Material Instabilities in Continuum Mechanics and Related Mathematical Problems (1988), Clarendon Press), 329-342 · Zbl 0632.76119 |
| [31] | Novick-Cohen, A.; Pego, R. L., Stable patterns in a viscous diffusion equation, Trans. Am. Math. Soc., 324, 331-351 (1991) · Zbl 0738.35035 |
| [32] | Padròn, V., Sobolev regularization of a nonlinear ill-posed parabolic problem as a model for aggregating populations, Commun. Partial Differ. Equ., 23, 457-486 (1998) · Zbl 0910.35138 |
| [33] | Pedregal, P., Parametrized Measures and Variational Principles, Progress in Nonlinear Differential Equations and Their Applications, vol. 30 (1997), Birkhäuser · Zbl 0879.49017 |
| [34] | Perona, P.; Malik, J., Scale space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12, 629-639 (1990) |
| [35] | Plotnikov, P. I., Passing to the limit with respect to viscosity in an equation with variable parabolicity direction, Differ. Equ., 30, 614-622 (1994) · Zbl 0824.35100 |
| [36] | Slemrod, M., Dynamics of measure valued solutions to a backward-forward heat equation, J. Dyn. Differ. Equ., 3, 1-28 (1991) · Zbl 0747.35013 |
| [37] | Smarrazzo, F., On a class of equations with variable parabolicity direction, Discrete Contin. Dyn. Syst., 22, 729-758 (2008) · Zbl 1156.35403 |
| [38] | Smarrazzo, F., On a class of quasilinear elliptic equations with degenerate coerciveness and measure data, Adv. Nonlinear Stud. (2017) |
| [39] | Smarrazzo, F.; Tesei, A., Degenerate regularization of forward-backward parabolic equations: the regularized problem, Arch. Ration. Mech. Anal., 204, 85-139 (2012) · Zbl 1255.35149 |
| [40] | Smarrazzo, F.; Tesei, A., Degenerate regularization of forward-backward parabolic equations: the vanishing viscosity limit, Math. Ann., 355, 551-584 (2013) · Zbl 1264.35275 |
| [41] | Tomassetti, G., Smooth and non-smooth regularizations of the nonlinear diffusion equation, Discrete Contin. Dyn. Syst., Ser. S, 10, 1519-1537 (2017) · Zbl 1366.35098 |
| [42] | Zhang, K., Existence of infinitely many solutions for the one-dimensional Perona-Malik model, Calc. Var. Partial Differ. Equ., 26, 171-199 (2006) · Zbl 1096.35067 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
