Share This Article:

Semilinear Venttsel’ Problems in Fractal Domains

Abstract Full-Text HTML Download Download as PDF (Size:1042KB) PP. 1820-1833
DOI: 10.4236/am.2014.512175    2,784 Downloads   3,531 Views   Citations


We study a semilinear parabolic problem with a semilinear dynamical boundary condition in an irregular domain with fractal boundary. Local existence, uniqueness and regularity results for the mild solution, are established via a semigroup approach. A sufficient condition on the initial datum for global existence is given.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Lancia, M. and Vernole, P. (2014) Semilinear Venttsel’ Problems in Fractal Domains. Applied Mathematics, 5, 1820-1833. doi: 10.4236/am.2014.512175.


[1] Beberns, J. and Eberly, D. (1989) Mathematical Problems from Combustion Theory. Applied Mathematical Sciences, 83, Springer Verlag, NewYork.
[2] Giga, Y. (1986) Solutions for Semilinear Parabolic Equations in and Regularity of Weak Solutions of the Navier Stokes System. Journal of Differential Equations, 62, 186-212.
[3] Venttsel, A.D. (1959) On Boundary Conditions for Multidimensional Diffusion Processes. Teoriya Veroyatnostei i ee Primeneniya, 4, 172-185, English Translation: Theory of Probability and Its Application, 4, 164-177.
[4] Coclite, G.M., Goldstein, G.R. and Goldstein, J.A. (2009) Stability of Parabolic Problems with Nonlinear Wentzell Boundary Conditions. Journal of Differential Equations, 246, 2434-2447.
[5] Evans, L.C. (1977) Regularity Properties for the Heat Equation Subject Non Linear Boundary Constraints. Nonlinear Analysis: Theory, Methods & Applications, 1, 593-602.
[6] Goldestein, R.G. (2006) Derivation and Physical Interpretation of General Boundary Conditions. Advances in Differential Equations, 11, 57-480.
[7] Favini, A., Goldestein, R.G. and Romanelli, S. (2002) The Heat Equation with Generalized Wentzell Boundary Condition. Journal of Evolution Equations, 2, 1-19.
[8] Lancia, M.R. and Vernole, P. (2012) Semilinear Evolution Transmission Problems across Fractal Layers. Nonlinear Analysis: Theory, Methods & Applications, 75, 4222-4240.
[9] Lancia, M.R. and Vernole, P. (2013) Semilinear Fractal Problems: Approximation and Regularity Results. Nonlinear Analysis: Theory, Methods & Applications, 80, 216-232.
[10] Lancia, M.R. and Vernole, P. (2014) Semilinear Evolution Problems with Ventcel-Type Condition on Fractal Boundaries. International Journal of Differential Equations, 2014, Article ID: 461046.
[11] Lancia, M.R. and Vernole, P. (2014) Venttsel’ Problems in Fractal Domains. Journal of Evolution Equations, Published Online.
[12] Warma, M. (2012) Regularity and Well-Posedness of Some Quasi-Linear Elliptic and Parabolic Problems with Nonlinear General Wentzell Boundary Conditions on Nonsmooth Domains. Nonlinear Analysis: Theory, Methods & Applications, 75, 5561-5588.
[13] Lunardi, A. (1995) Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, 16, Birk?uses Verlag, Basel.
[14] Cazenave, T. and Haraux A. (1998) An Introduction to Semilinear Evolution Equations. Oxford Science Publications, Oxford.
[15] Henry, D. (1981) Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin.
[16] Pazy, A. (1983) Semigroup of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, 44, Published Online.
[17] Tanabe, H. (1979) Equations of Evolution. Pitman, London.
[18] Weissler, F.B. (1980) Local Existence and Nonexistence of Semilinear Parabolic Equations in Lp. Indiana University Mathematics Journal, 29, 79-102.
[19] Kumagai, T. (2000) Brownian Motion Penetrating Fractals. Journal of Functional Analysis, 170, 69-92.
[20] Falconer, K. (1990) The Geometry of Fractal Sets. 2nd Edition, Cambridge Univ. Press, Cambridge.
[21] Jonsson, A. and Wallin, H. (1984) Function Spaces on Subset of Rn. Part 1, Mathematics Reports, 2, Harwood Academic Publishers, London.
[22] Freiberg, U. and Lancia, M. R. (2004) Energy Form on a Closed Fractal Curve. Zeitschrift für Analysis und ihre Anwendungen, 23, 115-135.
[23] Necas, J. (1967) Les mèthodes directes en thèorie des èquationes elliptiques. Masson, Paris.
[24] Adams D.R. and Hedberg D.R. (1966) Function Spaces and Potential Theory. Springer-Verlag, Berlin.
[25] Mosco, U. and Vivaldi, M.A. (2003) Variational Problems with Fractal Layers. Rendiconti della Accademia nazionale delle scienze detta dei XL.: Memorie di matematica e delle sue applicazioni, 27, 237-251.
[26] Triebel, H. (1997) Fractals and Spectra Related to Fourier Analysis and Function Spaces. Monographs in Mathematics, 91, Birkh?user, Basel.
[27] Fukushima, M., Oshima, Y. and Takeda, M. (1994) Dirichlet Forms and Symmetric Markov Processes. de Gruyter Studies in Mathematics, 19, W. de Gruyter, Berlin.
[28] Kato, T. (1977) Perturbation Theory for Linear Operators. 2nd Edition, Springer, Berlin.
[29] Dautray, R. and Lions, J.L. (1988) Mathematical Analysis and Numerical Methods for Science and Technology. 2, Springer-Verlag, Berlin.
[30] Davies, E.B. (1989) Heat Kernels and Spectral Theory. Cambridge Univ. Press, Cambridge.
[31] Fukushima, M. and Shima, T. (1992) On a Spectral Analysis for the Sierpinski Gasket. Potential Analysis, 1, 1-35.
[32] Rammal, R. and Tolouse G. (1983) Random Walks on Fractal Structures and Percolation Clusters. Journal de Physique Lettres, 44, 13-22.
[33] Kigami, J. (2001) Analysis on Fractals, Cambridge Tracts in Mathematics. 143, Cambridge University Press, Cambridge.
[34] Mosco, U. (1997) Variational Fractals, Dedicated to Ennio De Giorgi. Annali della Scuola Normale Superiore di Pisa, 25, 683-712.
[35] Bergh, J. and Löfström, J. (1976) Interpolation Spaces. Springer-Verlag, Berlin.
[36] Weissler, F.B. (1979) Semilinear Evolution Equations in Banach Spaces. Journal of Functional Analysis, 32, 277-296.
[37] Komatsu, H. (1966) Fractional Powers of Operators. Pacific Journal of Mathematics, 19, 285-346.
[38] Weissler, F.B. (1981) Existence and Non-Existence of Global Solutions for a Semilinear Heat Equation. Israel Journal of Mathematics, 38, 29-40.
[39] Lancia, M.R. and Vernole, P. (2006) Convergence Results for Parabolic Transmission Problems across Highly Conductive Layers with Small Capacity. Advances in Mathematical Sciences and Applications, 16, 411-445.
[40] Lancia, M.R. (2002) A Transmission Problem with a Fractal Interface. Zeitschrift für Analysis und ihre Anwendungen, 21, 113-133.
[41] Lancia, M.R. (2003) Second Order Transmission Problems across a Fractal Surface. Rendiconti della Accademia nazionale delle scienze detta dei XL.: Memorie di matematica e delle sue applicazioni, 27, 191-213.
[42] Lancia, M.R. and Vivaldi, M.A. (1999) Lipschitz Spaces and Besov Traces on Self-Similar Fractals. Rendiconti della Accademia nazionale delle scienze detta dei XL.: Memorie di matematica e delle sue applicazioni, 23, 101-106.
[43] Jerison, D. and Kening, C.E. (1982) Boundary Behaviour of Harmonic Functions in Nontangentially Accessible Domains. Advances in Mathematics, 46, 80-147.
[44] Nystrom, K. (1994) Smoothness Properties of Solutions to Dirichlet Problems in Domains with a Fractal Boundary. Doctoral Thesis, University of Umeä, Umeä.

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.