Semilinear Venttsel’ Problems in Fractal Domains


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.

Share and Cite:

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

Conflicts of Interest

The authors declare no conflicts of interest.


[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.

Copyright © 2023 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.