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Keywords:
Vectorial parabolic equation, quasilinear diffusion, dynamic boundary condition
Vectorial parabolic equation, quasilinear diffusion, dynamic boundary condition
Summary:
In this paper, we consider a class of initial-boundary value problems for quasilinear PDEs, subject to the dynamic boundary conditions. Each initial-boundary problem is denoted by (S)$_\varepsilon$ with a nonnegative constant $\varepsilon$, and for any $\varepsilon\ge0$, (S)$_\varepsilon$ can be regarded as a vectorial transmission system between the quasilinear equation in the spatial domain $\Omega$, and the parabolic equation on the boundary $\Gamma:=\partial \Omega$, having a sufficient smoothness. The objective of this study is to establish a mathematical method, which can enable us to handle the transmission systems of various vectorial mathematical models, such as the Bingham type flow equations, the Ginzburg– Landau type equations, and so on. On this basis, we set the goal of this paper to prove two Main Theorems, concerned with the well-posedness of (S)$_\varepsilon$(S) with the precise representation of solution, and $\varepsilon$-dependence of (S)$_\varepsilon$, for $\varepsilon \ge0$.
References:
[1] Attouch, H.: Variational Convergence for Functions and Operators. Applicable Mathematics Series. Pitman (Advanced Publishing Program), Boston, MA, 1984. MR 0773850
[2] Barbu, V.: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monographs in Mathematics. Springer, New York, 2010. MR 2582280
[3] Brézis, H.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). MR 0348562
[4] Colli, P., Gilardi, G., Nakayashiki, R., Shirakawa, K.: A class of quasi-linear Allen–Cahn type equations with dynamic boundary conditions. Nonlinear Anal., 158:32–59, 2017. DOI 10.1016/j.na.2017.03.020 | MR 3661429
[5] Giga, Y., Kashima, Y., Yamazaki, N.: Local solvability of a constrained gradient system of total variation. Abstr. Appl. Anal., (8):651–682, 2004. MR 2096945
[6] Ito, A., Yamazaki, N., Kenmochi, N.: Attractors of nonlinear evolution systems generated by time-dependent subdifferentials in Hilbert spaces. Discrete Contin. Dynam. Systems, (Added Volume I):327–349, 1998. Dynamical systems and differential equations, Vol. I (Springfield, MO, 1996). MR 1720614
[7] Kenmochi, N.: Solvability of nonlinear evolution equations with time-dependent constraints and applications. Bull. Fac. Education, Chiba Univ. http://ci.nii.ac.jp/naid/110004715232,30:1–87, 1981.
[8] Kenmochi, N.: Pseudomonotone operators and nonlinear elliptic boundary value problems. J. Math. Soc. Japan, 27:121–149, 1975. DOI 10.2969/jmsj/02710121 | MR 0372419
[9] Mosco, U.: Convergence of convex sets and of solutions of variational inequalities. Advances in Math., 3:510–585, 1969. DOI 10.1016/0001-8708(69)90009-7 | MR 0298508
[10] Nakayashiki, R., Shirakawa, K.: Weak formulation for singular diffusion equations with dynamic boundary condition. Springer INdAM Series. to appear, 2017. MR 3751650
[11] Savaré, G., Visintin, A.: Variational convergence of nonlinear diffusion equations: applications to concentrated capacity problems with change of phase. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 8(1):49–89, 1997. MR 1484545
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