×

Global existence of weak solutions for the nonlocal energy-weighted reaction-diffusion equations. (English) Zbl 1407.35111

The authors introduce and study the non-local energy-weighted reaction-diffusion equation \[ u_t = -E_\varepsilon^M(u) \nabla_X E_\varepsilon(u) \] where \[ E_\varepsilon(u) =\int_\Omega \left[\frac{\varepsilon |\nabla u|^2}{2}+\frac{W(u)}{\varepsilon}\right]\,dx \] for some double-well potential \(W\), \[ E_\varepsilon^M(u) = \min(E_\varepsilon(u), M) \] and \(\nabla_X\) is the gradient in \(X=L^2(\Omega)\).
By intricately adapting the Galerkin method to this modified reaction-diffusion setting, the authors show that there exists a weak solution \(u\in L^2(0,T,H_0^1(\Omega))\) for any initial conditions \(u_0\in H_0^1(\Omega)\) if Neumann boundary conditions are imposed at \(\partial \Omega\) and if \(\|f\|_{C^0}<\infty\) and \(\|f\|_{C^1}<\infty\) for \(f(u)=W'(u)\).

MSC:

35K57 Reaction-diffusion equations
35D30 Weak solutions to PDEs
35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
35A35 Theoretical approximation in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI Euclid

References:

[1] S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica 27 (1979), no. 6, 1085–1095.
[2] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. · Zbl 1090.35002
[3] K. A. Brakke, The Motion of a Surface by its Mean Curvature, Mathematical Notes 20, Princeton University Press, Princeton, N.J., 1978. · Zbl 0386.53047
[4] L. Bronsard and R. V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations 90 (1991), no. 2, 211–237. · Zbl 0735.35072
[5] X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations 96 (1992), no. 1, 116–141. · Zbl 0765.35024
[6] X. Chen and C. M. Elliott, Asymptotics for a parabolic double obstacle problem, Proc. Roy. Soc. London Ser. A 444 (1994), no. 1922, 429–445. · Zbl 0814.35044
[7] Y. G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), no. 3, 749–786. · Zbl 0696.35087
[8] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 1998. · Zbl 0902.35002
[9] L. C. Evans, H. M. Soner and P. E. Souganidis, Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math. 45 (1992), no. 9, 1097–1123. · Zbl 0801.35045
[10] L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differential Geom. 33 (1991), no. 3, 635–681. · Zbl 0726.53029
[11] P. C. Fife, Dynamics of Internal Layers and Diffusive Interfaces, CBMS-NSF Regional Conference Series in Applied Mathematics 53, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. · Zbl 0684.35001
[12] R. F. Gariepy and W. P. Ziemer, Modern Real Analysis, PWS Publishing Company, Boston, 1995.
[13] T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature, J. Differential Geom. 38 (1993), no. 2, 417–461. · Zbl 0784.53035
[14] G. Karali and T. Ricciardi, On the convergence of a fourth order evolution equation to the Allen-Cahn equation, Nonlinear Anal. 72 (2010), no. 11, 4271–4281. · Zbl 1187.35006
[15] S. Kondo and T. Miura, Reaction-diffusion model as a framework for understanding biological pattern formation, Science 329 (2010), no. 5999, 1616–1620. · Zbl 1226.35077
[16] P. de Mottoni and M. Schatzman, Evolution géométric d’interfaces, C. R. Acad. Paris Sci. Sér. I Math. 309 (1989), 453–458. · Zbl 0698.35078
[17] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations 26 (2001), no. 1-2, 101–174. · Zbl 0984.35089
[18] B. D. Reddy, Introductory Functional Analysis: With applications to boundary value problems and finite elements, Texts in Applied Mathematics 27, Springer-Verlag, New York, 1998. · Zbl 0893.46002
[19] J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA J. Appl. Math. 48 (1992), no. 3, 249–264. · Zbl 0763.35051
[20] J. Rubinstein, P. Steinberg and J. B. Keller, Fast reaction, slow diffusion, and curve shortening, SIAM J. Appl. Math. 49 (1989), no. 1, 116–133. · Zbl 0701.35012
[21] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. 33 (2013), no. 5, 2105–2137. · Zbl 1303.35121
[22] J. Simon, Compact sets in the space \(L^p(0,T;B)\), Ann. Mat. Pura Appl. (4) 146 (1987), 65–96. · Zbl 0629.46031
[23] A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B 237 (1952), no. 641, 37–72. · Zbl 1403.92034
[24] M. J. Ward, Metastable bubble solutions for the Allen-Cahn equation with mass conservation, SIAM J. Appl. Math. 56 (1996), no. 5, 1247–1279. · Zbl 0870.35011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.