## A three-point boundary value problem with an integral two-space-variables condition for parabolic equations.(English)Zbl 1124.35319

Summary: We study a three-point boundary value problem with an integral two-space-variables condition for a class of parabolic equations. The existence and uniqueness of the solution in the functional weighted Sobolev space are proved. The proof is based on two-sided a priori estimates and on the density of the range of the operator generated by the problem considered.

### MSC:

 35K20 Initial-boundary value problems for second-order parabolic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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### References:

 [1] Samarskii, A.A., Some problems in differential equations theory, Differ. uravn., 16, 11, 1221-1228, (1980) · Zbl 0519.35069 [2] Choi, Y.S.; Chan, K.Y., A parabolic equation with nonlocal boundary conditions arising from electrochemistry, Nonlinear anal., 18, 317-331, (1992) · Zbl 0757.35031 [3] Ewing, R.E.; Lin, T., A class of parameter estimation techniques for fluid flow in porous media, Adv. water ressources, 14, 89-97, (1991) [4] Shi, P., Weak solution to evolution problem with a nonlocal constraint, SIAM J. anal., 24, 46-58, (1993) · Zbl 0810.35033 [5] Kartynnik, A.V., Three-point boundary-value problem with an integral space-variable condition for a second-order parabolic equation, Differ. equ., 26, 1160-1166, (1990) · Zbl 0729.35053 [6] Batten, G.W., Second-order correct boundary conditions for the numerical solution of the mixed boundary problem for parabolic equations, Math. comput., 17, 405-413, (1963) · Zbl 0133.38601 [7] Bouziani, A.; Benouar, N.E., Mixed problem with integral conditions for a third order parabolic equation, Kobe J. math., 15, 47-58, (1998) · Zbl 0921.35068 [8] Cannon, J.R., The solution of the heat equation subject to the specification of energy, Quart. appl. math., 21, 155-160, (1963) · Zbl 0173.38404 [9] Cannon, J.R., The one-dimensional heat equation, () · Zbl 0168.36002 [10] Cannon, J.R.; Perez Esteva, S.; Van Der Hoek, J., A Galerkin procedure for the diffusion equation subject to the specification of mass, SIAM J. numer. anal., 24, 499-515, (1987) · Zbl 0677.65108 [11] Ionkin, N.I., Solution of a boundary-value problem in heat condition with a nonclassical boundary condition, Differ. uravn., 13, 294-304, (1977) · Zbl 0403.35043 [12] Kamynin, N.I., A boundary value problem in the theory of the heat condition with non classical boundary condition, U.S.S.R. comput. math. math. phys., 4, 33-59, (1964) [13] Shi, P.; Shillor, M., () [14] Marhoune, A.L.; Bouzit, M., High order differential equations with integral boundary condition, Far east J. math. sci., 18, 3, 341-450, (2005) · Zbl 1102.35329 [15] Denche, M.; Marhoune, A.L., Mixed problem with integral boundary condition for a high order mixed type partial differential equation, J. appl. math. stochastic anal., 16, 1, 69-79, (2003) · Zbl 1035.35085 [16] Denche, M.; Marhoune, A.L., Mixed problem with nonlocal boundary conditions for a third-order partial differential equation of mixed type, Int. J. math. math. sci., 26, 7, 417-426, (2001) · Zbl 1005.35004 [17] Denche, M.; Marhoune, A.L., High-order mixed type partial differential equations with integral boundary conditions, Electron. J. differ. equ., 2000, 60, 1-10, (2000) · Zbl 0967.35101 [18] Yurchuk, N.I., Mixed problem with an integral condition for certain parabolic equations, Differ. equ., 22, 1457-1463, (1986) · Zbl 0654.35041 [19] Kartynnik, A.V., Three-point boundary-value problem with an integral space-variable condition for a second-order parabolic equation, Differ. equ., 26, 9, 1160-1166, (1990) · Zbl 0729.35053 [20] Denche, M.; Marhoune, A.L., A three-point boundary value problem with an integral condition for parabolic equations with the Bessel operator, Appl. math. lett., 13, 85-89, (2000) · Zbl 0956.35072 [21] Hardy, G.H.; Littlewood, J.E.; Polya, G., Inequalities, (1934), Cambridge Press, Zbl 0010.10703 $$\mid$$ JFM 60.0169.01 · Zbl 0010.10703 [22] J. Prüss, G. Simonett, Maximal regularity for evolution equations in weighted Lp-spaces, Fachbereich Mathematik und Informatik, Martin-Luter-Universität Halle-Wittenberg, July 2002 [23] Hieber, M.; Prüss, J., Heat kernels and maximal lp – lq estimates for parabolic evolution equations, Comm. partial differ. equ., 22, 164761669, (1997) [24] Cannarasa, P.; Vespri, V., On maximal lp-regularity for abstract Cauchy problem, Boll. unione mat. italiana, 165-175, (1986)
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