# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2022040
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## Radon measure-valued solutions of unsteady filtration equations

 1 Dipartimento di Pianificazione, Design, Tecnologie dell'Architettura, Università Sapienza di Roma, via Flaminia 70, 00196 Roma, Italy 2 Università Campus Bio-Medico di Roma, Via Alvaro del Portillo 21, 00128 Roma, Italy 3 Dipartimento di Matematica "G. Castelnuovo", Università Sapienza di Roma, P.le A. Moro 5, I-00185 Roma 4 Istituto per le Applicazioni del Calcolo "M. Picone", CNR, Roma, Italy

*Corresponding author: Alberto Tesei

Received  November 2021 Revised  February 2022 Early access April 2022

We study Radon measure-valued solutions of a Cauchy-Dirichlet problem in one space dimension that describes nonlinear convection-diffusion phenomena in heterogeneous media. Well-posedness results are proven, which extend those previously known for the space homogeneous case.

Citation: Maria Michaela Porzio, Flavia Smarrazzo, Alberto Tesei. Radon measure-valued solutions of unsteady filtration equations. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022040
##### References:
 [1] M. Bertsch and R. Dal Passo, A parabolic equation with a mean-curvature type operator, In Nonlinear Diffusion Equations and Their Equilibrium States, 3, (Gregynog, 1989), Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 7 (1992), 89–97. [2] M. Bertsch, R. Dal Passo and B. Franchi, A degenerate parabolic problem in noncylindrical domains, Math. Ann., 294 (1992), 551-578.  doi: 10.1007/BF01934341. [3] H. Brezis and A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl., 62 (1983), 73-97. [4] C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 3$^{rd}$ edition, Springer, 2004. [5] J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4. [6] W.-M. Ni, The Mathematics of Diffusion, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972. [7] O. A. Oleinik, On equations of the unsteady filtration type, Dokl. Akad. Nauk SSSR, 113 (1957), 1210-1213. [8] M. Pierre, Nonlinear fast diffusion with measures as data, In Nonlinear Parabolic Equations: Qualitative Properties of Solutions, (Rome, 1985), Pitman Res. Notes Math. Ser. 149 (1987), 179–188. [9] M. Pierre, Uniqueness of the solutions of $\partial_t u - \Delta \zeta(u) = 0$ with initial datum a measure, Nonlin. Anal., 6 (1982), 175-187.  doi: 10.1016/0362-546X(82)90086-4. [10] M. M. Porzio, F. Smarrazzo and A. Tesei, Radon measure-valued solutions for a class of quasilinear parabolic equations, Arch. Rat. Mech. Anal., 210 (2013), 713-772.  doi: 10.1007/s00205-013-0666-0. [11] M. M. Porzio, Sm arrazzo and A. Tesei, Radon measure-valued solutions of nonlinear strongly degenerate parabolic equations, Calc. Var. Partial Differential Equations, 51 (2014), 401-437.  doi: 10.1007/s00526-013-0680-y. [12] M. M. Porzio, F. Smarrazzo and A. Tesei, Noncoercive diffusion equations with Radon measures as initial data, J. London Math. Soc., (2022).  doi: 10.1112/jlms.12548. [13] M. Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc., 51 (1984), 224 pp. doi: 10.1090/memo/0307. [14] M. Valadier, A course on Young measures, Rend. Ist. Mat. Univ. Trieste, 26 (1994), 349-394. [15] J. L. Vázquez, Porous Medium Equation. Mathematical Theory, Oxford University Press, Oxford, 2007. [16] J. L. Vazquez, Measure-valued solutions and phenomenon of blow-down in logarithmic diffusion, J. Math. Anal. Appl., 352 (2009), 515-547.  doi: 10.1016/j.jmaa.2008.06.032.

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##### References:
 [1] M. Bertsch and R. Dal Passo, A parabolic equation with a mean-curvature type operator, In Nonlinear Diffusion Equations and Their Equilibrium States, 3, (Gregynog, 1989), Progr. Nonlinear Differential Equations Appl., Birkhäuser Boston, Boston, MA, 7 (1992), 89–97. [2] M. Bertsch, R. Dal Passo and B. Franchi, A degenerate parabolic problem in noncylindrical domains, Math. Ann., 294 (1992), 551-578.  doi: 10.1007/BF01934341. [3] H. Brezis and A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl., 62 (1983), 73-97. [4] C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, 3$^{rd}$ edition, Springer, 2004. [5] J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4. [6] W.-M. Ni, The Mathematics of Diffusion, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972. [7] O. A. Oleinik, On equations of the unsteady filtration type, Dokl. Akad. Nauk SSSR, 113 (1957), 1210-1213. [8] M. Pierre, Nonlinear fast diffusion with measures as data, In Nonlinear Parabolic Equations: Qualitative Properties of Solutions, (Rome, 1985), Pitman Res. Notes Math. Ser. 149 (1987), 179–188. [9] M. Pierre, Uniqueness of the solutions of $\partial_t u - \Delta \zeta(u) = 0$ with initial datum a measure, Nonlin. Anal., 6 (1982), 175-187.  doi: 10.1016/0362-546X(82)90086-4. [10] M. M. Porzio, F. Smarrazzo and A. Tesei, Radon measure-valued solutions for a class of quasilinear parabolic equations, Arch. Rat. Mech. Anal., 210 (2013), 713-772.  doi: 10.1007/s00205-013-0666-0. [11] M. M. Porzio, Sm arrazzo and A. Tesei, Radon measure-valued solutions of nonlinear strongly degenerate parabolic equations, Calc. Var. Partial Differential Equations, 51 (2014), 401-437.  doi: 10.1007/s00526-013-0680-y. [12] M. M. Porzio, F. Smarrazzo and A. Tesei, Noncoercive diffusion equations with Radon measures as initial data, J. London Math. Soc., (2022).  doi: 10.1112/jlms.12548. [13] M. Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc., 51 (1984), 224 pp. doi: 10.1090/memo/0307. [14] M. Valadier, A course on Young measures, Rend. Ist. Mat. Univ. Trieste, 26 (1994), 349-394. [15] J. L. Vázquez, Porous Medium Equation. Mathematical Theory, Oxford University Press, Oxford, 2007. [16] J. L. Vazquez, Measure-valued solutions and phenomenon of blow-down in logarithmic diffusion, J. Math. Anal. Appl., 352 (2009), 515-547.  doi: 10.1016/j.jmaa.2008.06.032.
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