Existence to nonlinear parabolic problems with unbounded weights

  • Iwona ChlebickaEmail author
  • Anna Zatorska-Goldstein


We consider the weighted parabolic problem of the type
$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} u_t-\mathrm{div}(\omega _2(x)|\nabla u|^{p-2} \nabla u )= \lambda \omega _1(x) |u|^{p-2}u,&{} x\in \Omega ,\\ u(x,0)=f(x),&{} x\in \Omega ,\\ u(x,t)=0,&{} x\in \partial \Omega ,\ t>0,\\ \end{array}\right. \end{aligned} \end{aligned}$$
for quite a general class of possibly unbounded weights \( \omega _1,\omega _2\) satisfying the Hardy-type inequality. We prove existence of a global weak solution in the weighted Sobolev spaces provided that \(\lambda >0\) is smaller than the optimal constant in the inequality. The domain is assumed to be bounded or quasibounded. The obtained solution is proven to belong to
$$\begin{aligned} L^p({{\mathbb {R}}}_+; W_{(\omega _1,\omega _2),0}^{1,p}(\Omega ))\cap L^\infty ({{\mathbb {R}}}_+; L^2(\Omega )). \end{aligned}$$


Existence of solutions Hardy inequalities Parabolic problems Weighted p-Laplacian Weighted Sobolev spaces 

Mathematics Subject Classification

35K55 35A01 47J35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    B. Abdellaoui, I. Peral, and M. Walias. Some existence and regularity results for porous media and fast diffusion equations with a gradient term. Trans. Amer. Math. Soc., 367(7):4757–4791, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    C. T. Anh and T. D. Ke. On quasilinear parabolic equations involving weighted \(p\)-Laplacian operators. NoDEA Nonlinear Differential Equations Appl., 17(2):195–212, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    A. Attar, S. Merchán, and I. Peral. A remark on the existence properties of a semilinear heat equation involving a Hardy-Leray potential. J. Evol. Equ., 15(1):239–250, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    P. Baras and J. Goldstein. The heat equation with a singular potential. Trans. Amer. Math. Soc., 284(1):121–139, 1984.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    L. Boccardo, A. Dall’Aglio, T. Gallouët, and L. Orsina. Nonlinear parabolic equations with measure data. J. Funct. Anal., 147(1):237–258, 1997.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    L. Boccardo, T. Gallouët, and L. Orsina. Existence and nonexistence of solutions for some nonlinear elliptic equations. J. Anal. Math., 73:203–223, 1997.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    L. Boccardo and F. Murat. Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal., 19(6):581–597, 1992.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    L. Boccardo, F. Murat, and J.-P. Puel. Existence of bounded solutions for nonlinear elliptic unilateral problems. Ann. Mat. Pura Appl. (4), 152:183–196, 1988.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M. Bonforte, J. Dolbeault, M. Muratori, and B. Nazaret. Weighted fast diffusion equations (part i): Sharp asymptotic rates without symmetry and symmetry breaking in caffarelli-kohn-nirenberg inequalities. To Appear in Kinet. Rel. Mod., 2016.Google Scholar
  10. 10.
    M. Bonforte, J. Dolbeault, M. Muratori, and B. Nazaret. Weighted fast diffusion equations (part ii): Sharp asymptotic rates of convergence in relative error by entropy methods. To Appear in Kinet. Rel. Mod., 2016.Google Scholar
  11. 11.
    H. Brézis and E. Lieb. A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc., 88(3):486–490, 1983.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    A. Dall’Aglio, D. Giachetti, and I. Peral. Results on parabolic equations related to some Caffarelli-Kohn-Nirenberg inequalities. SIAM J. Math. Anal., 36(3):691–716, 2004/05.Google Scholar
  13. 13.
    B. Franchi, R. Serapioni, and F. Serra Cassano. Approximation and imbedding theorems for weighted Sobolev spaces associated with Lipschitz continuous vector fields. Boll. Un. Mat. Ital. B (7), 11(1):83–117, 1997.Google Scholar
  14. 14.
    J. P. García Azorero and I. Peral Alonso. Hardy inequalities and some critical elliptic and parabolic problems. J. Differential Equations, 144(2):441–476, 1998.Google Scholar
  15. 15.
    J. A. Goldstein, D. Hauer, and A. Rhandi. Existence and nonexistence of positive solutions of \(p\)-Kolmogorov equations perturbed by a Hardy potential. Nonlinear Anal., 131:121–154, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    A. Kufner and B. Opic. How to define reasonably weighted Sobolev spaces. Comment. Math. Univ. Carolin., 25(3):537–554, 1984.MathSciNetzbMATHGoogle Scholar
  17. 17.
    S. Merchán, L. Montoro, I. Peral, and B. Sciunzi. Existence and qualitative properties of solutions to a quasilinear elliptic equation involving the Hardy-Leray potential. Ann. Inst. H. Poincaré Anal. Non Linéaire, 31(1):1–22, 2014.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    B. Opic and A. Kufner. Hardy-type inequalities, volume 219 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow, 1990.Google Scholar
  19. 19.
    J. Simon. Compact sets in the space \(L^p(0,T;B)\). Ann. Mat. Pura Appl. (4), 146:65–96, 1987.zbMATHGoogle Scholar
  20. 20.
    I. Skrzypczak. Hardy-type inequalities derived from \(p\)-harmonic problems. Nonlinear Anal., 93:30–50, 2013.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    I. Skrzypczak and A. Zatorska-Goldstein. Existence of solutions to a nonlinear parabolic problem with two weights. to appear in Colloq. Math., 2018.Google Scholar
  22. 22.
    J.-L. Vazquez and E. Zuazua. The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential. J. Funct. Anal., 173(1):103–153, 2000.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Institute of MathematicsPolish Academy of SciencesWarsawPoland
  2. 2.Institute of Applied Mathematics and MechanicsUniversity of WarsawWarsawPoland

Personalised recommendations