# American Institute of Mathematical Sciences

December  2007, 6(4): 1051-1074. doi: 10.3934/cpaa.2007.6.1051

## $L^1$-estimates for the higher-order derivatives of solutions to parabolic equations subject to initial values of bounded total variation

 1 Sobolev Institute of Mathematics, 4, Acad. Koptyug prosp., 630090 Novosibirsk, Russian Federation 2 Dipartimento di Matematica, Università “Roma Tre”, 1, Largo S. L. Murialdo, 00146 Rome, Italy

Received  January 2007 Revised  June 2007 Published  September 2007

$L^1$-estimates are established for the higher-order derivatives of classical solutions to the homogeneous Cauchy problem for linear second-order one-dimensional parabolic equations of general form. It is required that the initial data is uniformly continuous and of bounded total variation on some given bounded interval. If the latter condition holds on every bounded interval, then uniform $L^1$-estimates can be proved for the higher-order derivatives. In contrast to earlier findings, where the case of bounded initial data with a continuity modulus satisfying a Dini condition was considered, no constraints are imposed to such a continuity modulus in this paper. In particular, the initial data are allowed to be unbounded. Sets of initial data, in general discontinuous, are also considered.
Citation: Denis R. Akhmetov, Renato Spigler. $L^1$-estimates for the higher-order derivatives of solutions to parabolic equations subject to initial values of bounded total variation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1051-1074. doi: 10.3934/cpaa.2007.6.1051
 [1] Yohei Fujishima. On the effect of higher order derivatives of initial data on the blow-up set for a semilinear heat equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 449-475. doi: 10.3934/cpaa.2018025 [2] Francis Ribaud. Semilinear parabolic equations with distributions as initial data. Discrete & Continuous Dynamical Systems, 1997, 3 (3) : 305-316. doi: 10.3934/dcds.1997.3.305 [3] Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the heat equation with concave-convex nonlinearity and initial data in weak-$L^p$ spaces. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1715-1732. doi: 10.3934/cpaa.2011.10.1715 [4] Joana Terra, Noemi Wolanski. Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 581-605. doi: 10.3934/dcds.2011.31.581 [5] Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365 [6] Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617 [7] Luz de Teresa, Enrique Zuazua. Identification of the class of initial data for the insensitizing control of the heat equation. Communications on Pure & Applied Analysis, 2009, 8 (1) : 457-471. doi: 10.3934/cpaa.2009.8.457 [8] Yaobin Ou, Pan Shi. Global classical solutions to the free boundary problem of planar full magnetohydrodynamic equations with large initial data. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 537-567. doi: 10.3934/dcdsb.2017026 [9] Bingkang Huang, Lan Zhang. A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data. Kinetic & Related Models, 2019, 12 (2) : 357-396. doi: 10.3934/krm.2019016 [10] Zhi-Qiang Shao. Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities. Communications on Pure & Applied Analysis, 2015, 14 (3) : 759-792. doi: 10.3934/cpaa.2015.14.759 [11] Kazuhiro Ishige. On the existence of solutions of the Cauchy problem for porous medium equations with radon measure as initial data. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 521-546. doi: 10.3934/dcds.1995.1.521 [12] Woocheol Choi, Youngwoo Koh. On the splitting method for the nonlinear Schrödinger equation with initial data in $H^1$. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3837-3867. doi: 10.3934/dcds.2021019 [13] Fengbai Li, Feng Rong. Decay of solutions to fractal parabolic conservation laws with large initial data. Communications on Pure & Applied Analysis, 2013, 12 (2) : 973-984. doi: 10.3934/cpaa.2013.12.973 [14] Kai-Seng Chou, Ying-Chuen Kwong. General initial data for a class of parabolic equations including the curve shortening problem. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2963-2986. doi: 10.3934/dcds.2020157 [15] Niklas Sapountzoglou, Aleksandra Zimmermann. Well-posedness of renormalized solutions for a stochastic $p$-Laplace equation with $L^1$-initial data. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2341-2376. doi: 10.3934/dcds.2020367 [16] Brian Smith and Gilbert Weinstein. On the connectedness of the space of initial data for the Einstein equations. Electronic Research Announcements, 2000, 6: 52-63. [17] Seung-Yeal Ha, Bingkang Huang, Qinghua Xiao, Xiongtao Zhang. A global existence of classical solutions to the two-dimensional kinetic-fluid model for flocking with large initial data. Communications on Pure & Applied Analysis, 2020, 19 (2) : 835-882. doi: 10.3934/cpaa.2020039 [18] Xiaoqiang Dai, Shaohua Chen. Global well-posedness for the Cauchy problem of generalized Boussinesq equations in the control problem regarding initial data. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021114 [19] Hongyong Cui, Peter E. Kloeden, Wenqiang Zhao. Strong $(L^2,L^\gamma\cap H_0^1)$-continuity in initial data of nonlinear reaction-diffusion equation in any space dimension. Electronic Research Archive, 2020, 28 (3) : 1357-1374. doi: 10.3934/era.2020072 [20] Young-Sam Kwon, Antonin Novotny. Derivation of geostrophic equations as a rigorous limit of compressible rotating and heat conducting fluids with the general initial data. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 395-421. doi: 10.3934/dcds.2020015

2020 Impact Factor: 1.916