Rocky Mountain Journal of Mathematics

The heat equation for local Dirichlet forms: Existence and blow up of nonnegative solutions

Tarek Kenzizi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We establish conditions ensuring either existence or blow up of nonnegative solutions for the following parabolic problem: \begin{equation} \begin{cases} Hu -Vu + ({\partial u}/{\partial t}) =0 & \mbox {in } X\times (0,T), \\ u(x,0)=u_{0}(x) & \mbox {in } X, \end{cases} \end{equation} where $T>0$, $X$ is a locally compact separable metric space, $H$ is a selfadjoint operator associated with a regular Dirichlet form $\mathcal E$; the initial value $u_{0}\in L^{2}(X,m)$, where $m$ is a positive Radon measure on Borel subset $U$ of $X$ such that $m(U)>0$ and $V$ is a Borel locally integrable function on $X$.

Article information

Source
Rocky Mountain J. Math., Volume 48, Number 8 (2018), 2573-2593.

Dates
First available in Project Euclid: 30 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1546138822

Digital Object Identifier
doi:10.1216/RMJ-2018-48-8-2573

Mathematical Reviews number (MathSciNet)
MR3894994

Zentralblatt MATH identifier
1403.35120

Subjects
Primary: 34B27: Green functions 35K08: Heat kernel 35K55: Nonlinear parabolic equations

Keywords
Dirichlet forms heat equation parabolic problem

Citation

Kenzizi, Tarek. The heat equation for local Dirichlet forms: Existence and blow up of nonnegative solutions. Rocky Mountain J. Math. 48 (2018), no. 8, 2573--2593. doi:10.1216/RMJ-2018-48-8-2573. https://projecteuclid.org/euclid.rmjm/1546138822


Export citation

References

  • S. Albeverio, J. Brasche and M. Rockner, Dirichlet forms and generalized Schrödinger operators, Lect. Notes Phys. 345 (1989), 1–42.
  • S. Albeverio, F. Gesztesy, W. Karwowski and L. Streit, On the connection between Schrodinger and Dirichlet forms, J. Math. Phys. 26 (1985), 2546–2553.
  • S. Albeverio and R. Hoegh-Krohn, Some remarks on Dirichlet forms and their applications to quantum mechanics and statistical mechanics, in Functional analysis and Markov processes, Springer, Berlin, 1982.
  • S. Albeverio, S. Kusuoka and L. Streit, Convergence of Dirichlet forms and associated Schrodinger operators, J. Funct. Anal. 68 (1986), 130–148.
  • S. Albeverio and Z. Ma, Perturbation of Dirichlet forms–Lower semiboundedness, closability, and form cores, J. Funct. Anal. 99 (1991), 332–356.
  • ––––, Diffusion processes with singular Dirichlet forms, in Stochastic analysis and applications, Progr. Probab. 26 (1991), 11–28.
  • A.B. Amor and T. Kenzizi, The heat equation for the Dirichlet fractional Laplacian with negative potentials: Existence and blow up of nonnegative solutions, Acta Math. Sinica (2017).
  • P. Baras and J.A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc. 284 (1984), 121–139.
  • A. Beldi, N.B. Rhouma and A.B. Amor, Pointwise estimate for the ground states of some classes of positivity preserving operators, Osaka J. Math. 50 (2013), 765–793.
  • P. Blanchard and Z. Ma, Semigroup of Schrödinger operators with potentials given by radon measures, in Stochastic processes, Physics and geometry, S. Albeverio, et al., eds., World Scientific, Singapore, 1990.
  • N. Bouleau and F. Hirsch, Dirichlet forms and analysis on Wiener space, de Gruyter Stud. Math. 14 (1991).
  • F. Brasche, Non-local point interactions and associated Dirichlet forms, Technical report, University of Bielefeld, FRG, Germany, 1986.
  • ––––, Dirichlet forms and nonstandard Schrodinger operators, Standard and nonstandard, World Scientific, Teaneck, NJ, 1989.
  • ––––, Generalized Schrodinger operators, An inverse problem in spectral analysis and the Efimov effect, in Stochastic processes, physics and geometry, World Scientific, Teaneck, NJ, 1990.
  • ––––, On the spectral properties of singular perturbed operators, in Dirichlet forms and stochastic processes, World Scientific, Teaneck, NJ, 1995.
  • J. Brasche, P. Exner, Y. Kuperin and P. Seha, Schrodinger operators with singular interactions, SFB 237 132 (1991), Bochum.
  • E.B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrodinger operators and Dirichlet Laplacians, J. Funct. Anal. 59 (1984), 335–395.
  • M. Fukushima, Y. Oshima and M. Takeda, Dirichlet forms and symmetric Markov processes, de Gruyter Stud. Math. 19 (1994).
  • J.A. Goldstein and Q.S. Zhang, Linear parabolic equations with strong singular potentials, Trans. Amer. Math. Soc. 129 (2003), 197–211.
  • A. Grigor'yan and L. Saloff-Coste, Heat kernel on manifolds with ends, Ann. Inst Fourier, Grenoble, 59 (2009), 1917–1997.
  • M. Hino, Some properties of enrergy measures on Sierpinski gasket type fractals, arXiv: 1510.00475v1.
  • M. Hino and K. Nakahara, On singularity of energy measures on self similar sets, II, Bull. Lond. Math. Soc. 38 (2006), 1019–1032.
  • T. Kenzizi, Existence of nonnegative solutions for parabolic problem on Dirichlet forms, Appl. Anal. (2017).
  • O.A. Ladyzenskaja, V.A. Solonnikov and N.N. Uralceva, Linear and quasilinear equations of parabolic type, Trans. Math. Mono. 23 (1967).
  • D. Lenz, P. Stolmann and I. Veselić, The Allegretto-Piepenbrink theorem for strongly local Dirichlet forms, Doc. Math. (2009).
  • P. Li and S.T. Yau, On the parabolic kernel of the Schrodinger operator, Acta Math. 156 (1986), 153–201.
  • J. Lier and L. Saloff-Coste, Parabolic Harnack inequality for time-dependent non-symmetric Dirichlet form, arXiv: 1205.6493v5.
  • P. Stolmann and J. Voigt, Perturbation of Dirichlet forms by measures, Potent. Anal. \bf5 (1996), 109–138.
  • K.T. Sturm, Analysis on local Dirichlet spaces, II, Gaussian upper bounds for the fundamental solutions of parabolic equations, Osaka J. Math. \bf32 (1995), 275–312.
  • ––––, Analysis on local Dirichlet spaces, I, Recurrence, conservativeness and $L^{p}$-Liouville properties, J. reine angew. Math. 456 (1994), 173–196.
  • ––––, Analysis on local Dirichlet spaces, III, The parabolic Harnack inequality, Osaka J. Math. \bf75 (1996), 273–297.
  • ––––, Schrödinger operators with arbitrary nonnegative potentials, in Operator calculus and spectral theory, Birkhauser, Berlin, 1992.
  • ––––, When does a Schrödinger heat equation permit positive solutions?, World Scientific, Singapore, 2010.
  • C. Xavier and Y. Martel, Existence versus explosion instantanée pour des equations de la chaleur linéaires avec potentiel singulier, C.R. Acad. Sci. Paris Math. 329 (1999), 973–978.