Geodesic
Stabilization of the solutions of the second boundary value problem for a second order parabolic equation
Sbornik. Mathematics, Tome 30 (1976) no. 4, pp. 403-440 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper is a continuation of work (RZhMat., 1973, 10B301) in which in the case of a “noncontracting” unbounded domain $\Omega$ there is distinguished a geometric characteristic $v(R)=\operatorname{mes}(\Omega\cap\{|x| of the domain $\Omega$ that determines (under the fulfillment of a certain condition of “regularity” of the domain) the rate of stabilization for $t\to\infty$ of the solution in $(t>0)\times\Omega$ of the following second boundary value problem for a parabolic equation: $$ u_t=\sum_{i,j=1}^n\bigl(a_{i,j}(t,x)u_{x_i}\bigr)_{x_j},\qquad\frac{\partial u}{\partial N}\Bigr|_{x\in\partial\Omega}=0,\quad u|_{t=0}=\varphi(x) $$ in which the initial function $\varphi(x)$ decreases sufficiently rapidly as $|x|\to\infty$. It is proved in the present paper that the same characteristic also determines the rate of stabilization of the solution in a class of “contracting” ($\lim_{R\to\infty}v(R)/R=0$) domains $\Omega$. In this case, as in the case of a “noncontracting” domain, $\|u(t,x)\|_{L_\infty(\Omega)}$ tends to zero as $t\to\infty$ like $1/v(\sqrt{t})$: there exist estimates of the function $\|u(t,x)\|_{L_\infty(\Omega)}$ from above and from below having such an order of decrease. Bibliography: 11 titles. 
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A. K. Gushchin. Stabilization of the solutions of the second boundary value problem for a second order parabolic equation. Sbornik. Mathematics, Tome 30 (1976) no. 4, pp. 403-440. http://geodesic.mathdoc.fr/item/SM_1976_30_4_a0/

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[2] A. K. Guschin, “Ob otsenkakh reshenii kraevykh zadach dlya parabolicheskogo uravneniya vtorogo poryadka”, Trudy Matem. in-ta AN SSSR im. V. A. Steklova, CXXVI (1973), 5–45

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[4] A. K. Guschin, “Nekotorye svoistva obobschennogo resheniya vtoroi kraevoi zadachi dlya parabolicheskogo uravneniya”, Matem. sb., 97(139) (1975), 242–261 | Zbl

[5] A. K. Guschin, “O skorosti stabilizatsii resheniya kraevoi zadachi dlya parabolicheskogo uravneniya”, Sib. matem. zh., X:1 (1969), 43–57

[6] J. Nash, “Continuity of solutions of parabolic and elliptic equations”, Amer. J. Math., 80 (1958), 931–953 ; Matematika, 4:1 (1960), 31–52 | DOI | MR

[7] A. K. Guschin, “O skorosti stabilizatsii resheniya parabolicheskogo uravneniya v neogranichennoi oblasti”, Diff. uravneniya, VI:4 (1970), 741–761

[8] A. K. Guschin, “Ob otsenkakh reshenii II i III kraevykh zadach dlya parabolicheskogo uravneniya vtorogo poryadka v neogranichennykh po prostranstvennym peremennym oblastyakh”, DAN SSSR, 206:4 (1972), 788–791 | Zbl

[9] A. K. Guschin, “Ob otsenkakh snizu reshenii vtoroi kraevoi zadachi dlya parabolicheskogo uravneniya vtorogo poryadka v neogranichennykh po prostranstvennym peremennym oblastyakh”, DAN SSSR, 206:6 (1972), 1284–1287 | Zbl

[10] A. K. Guschin, Otsenki reshenii vtoroi i tretei kraevykh zadach v neogranichennoi po prostranstvennym peremennym oblasti dlya parabolicheskogo uravneniya vtorogo poryadka, Doktorskaya dissertatsiya, Moskva, 1973

[11] A. K. Guschin, “Ob otsenke integrala Dirikhle v neogranichennykh oblastyakh”, Matem. sb., 99 (141) (1976), 282–294 | Zbl