Second-order time degenerate parabolic equations

Author:
Margaret C. Waid

Journal:
Trans. Amer. Math. Soc. **170** (1972), 31-55

MSC:
Primary 35K10

DOI:
https://doi.org/10.1090/S0002-9947-1972-0304860-1

MathSciNet review:
0304860

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the degenerate parabolic operator $Lu = \sum \nolimits _{i,j = 1}^n {{a^{ij}}{u_{{x_j}{x_j}}}} + \sum \nolimits _{i = 1}^n {{b^i}{u_{{x_i}}}} - c{u_t} + du$ where the coefficients of $L$ are bounded, real-valued functions defined on a domain $D = \Omega \times (0,T] \subset {R^{n + 1}}$. Classically, $c(x,t) \equiv 1$ or, equivalently, $c(x,t) \geq \eta > 0$ for all $(x,t) \in \bar D$. We assume only that $c$ is non-negative. We prove weak maximum principles and Harnack inequalities. Assume that ${a^{ij}}$ is constant, the coefficients of $L$ and $f$ and their derivatives with respect to time are uniformly Hölder continuous (exponent $\alpha$) in $\bar D,\bar D$ has sufficiently nice boundary, $c > 0$ on the normal boundary of $D$, $\psi \in {\bar C_{z + \alpha }}$, and $L\psi = f$ on $\partial B = \partial (\bar D \cap \{ t = 0\} )$. Then there exists a unique solution $u$ of the first initial-boundary value problem $Lu = f,u = \psi$ on $\bar B + (\partial B \times [0,T])$; and, furthermore, $u \in {\bar C_{2 + \alpha }}$. All results require proofs that differ substantially from the classical ones.

- D. G. Aronson,
*Non-negative solutions of linear parabolic equations*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)**22**(1968), 607–694. MR**435594** - D. G. Aronson,
*Regularity propeties of flows through porous media*, SIAM J. Appl. Math.**17**(1969), 461–467. MR**247303**, DOI https://doi.org/10.1137/0117045 - D. G. Aronson and James Serrin,
*Local behavior of solutions of quasilinear parabolic equations*, Arch. Rational Mech. Anal.**25**(1967), 81–122. MR**244638**, DOI https://doi.org/10.1007/BF00281291
W. T. Ford, - Wayne T. Ford,
*The first initial-boundary value problem for a nonuniform parabolic equation*, J. Math. Anal. Appl.**40**(1972), 131–137. MR**320532**, DOI https://doi.org/10.1016/0022-247X%2872%2990035-2 - Avner Friedman,
*Partial differential equations of parabolic type*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR**0181836** - C. Denson Hill,
*A sharp maximum principle for degenerate elliptic-parabolic equations*, Indiana Univ. Math. J.**20**(1970/71), 213–229. MR**287175**, DOI https://doi.org/10.1512/iumj.1970.20.20020 - J. J. Kohn and L. Nirenberg,
*Degenerate elliptic-parabolic equations of second order*, Comm. Pure Appl. Math.**20**(1967), 797–872. MR**234118**, DOI https://doi.org/10.1002/cpa.3160200410 - O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva,
*Lineĭ nye i kvazilineĭ nye uravneniya parabolicheskogo tipa*, Izdat. “Nauka”, Moscow, 1967 (Russian). MR**0241821** - O. A. Ladyženskaja and N. N. Ural′ceva,
*On the Hölder continuity of the solutions and the derivatives of linear and quasi-linear equations of elliptic and parabolic types*, Trudy Mat. Inst. Steklov.**73**(1964), 172–220 (Russian). MR**0173855**
E. E. Levi, - Jürgen Moser,
*A Harnack inequality for parabolic differential equations*, Comm. Pure Appl. Math.**17**(1964), 101–134. MR**159139**, DOI https://doi.org/10.1002/cpa.3160170106 - Olga Oleĭnik,
*Quasi-linear second-order parabolic equations with many independent variables*, Seminari 1962/63 Anal. Alg. Geom. e Topol., Vol. 1, Ist. Naz. Alta Mat., Ediz. Cremonese, Rome, 1965, pp. 332–354. MR**0194763** - O. A. Oleĭnik and S. N. Kružkov,
*Quasi-linear parabolic second-order equations with several independent variables*, Uspehi Mat. Nauk**16**(1961), no. 5 (101), 115–155 (Russian). MR**0141892** - Murray H. Protter and Hans F. Weinberger,
*Maximum principles in differential equations*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR**0219861** - Neil S. Trudinger,
*Pointwise estimates and quasilinear parabolic equations*, Comm. Pure Appl. Math.**21**(1968), 205–226. MR**226168**, DOI https://doi.org/10.1002/cpa.3160210302
E. D. Williams,

*Elements of simulation of fluid flow in porous media*, Texas Tech University Press, Lubbock, Tex., 1971.

*Sulle equazioni lineari totalmente ellittiche alle derivate parziali*, Rend. Circ. Mat. Palermo

**24**(1907), 275-317.

*The numerical solution of degenerate parabolic equations*, Doctoral Dissertation, Texas Tech University, Lubbock, Tex., 1971.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
35K10

Retrieve articles in all journals with MSC: 35K10

Additional Information

Keywords:
Degenerate parabolic equations,
nonuniformly parabolic operator,
first initial-boundary value problem,
maximum principles,
Harnack inequality,
a priori estimates,
existence of a unique classical solution

Article copyright:
© Copyright 1972
American Mathematical Society