Geodesic
Exact Non-Self-Similar Solutions of the Equation
Matematičeskie zametki, Tome 70 (2001) no. 5, pp. 787-792.

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In this paper, we obtain new exact non-self-similar solutions of the nonlinear diffusion equation $$ u_t=\Delta \ln u,\quad u\triangleq u(\mathbf x,t):\Omega\times\mathbb R^+\to\mathbb R, \quad\mathbf x\in\mathbb R^n, $$ where $\Omega\subset\mathbb R^n$ is the domain and $\mathbb R^+=\{t:0\le t+\infty\}$, $u(\mathbf x,t)\ge0$ is the temperature of the medium. 
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È. I. Semenov; G. A. Rudykh. Exact Non-Self-Similar Solutions of the Equation. Matematičeskie zametki, Tome 70 (2001) no. 5, pp. 787-792. http://geodesic.mathdoc.fr/item/MZM_2001_70_5_a13/

[1] Galaktionov V. A., Dorodnitsyn V. A., Elenin G. G., Kurdyumov S. P., Samarskii A. A., “Kvazilineinoe uravnenie teploprovodnosti: obostrenie, lokalizatsiya, simmetriya, tochnye resheniya, asimptotiki, struktury”, Itogi nauki i tekhniki. Sovremennye problemy matematiki. Noveishie dostizheniya, 28, VINITI, M., 1986, 95–205 | MR

[2] Samarskii A. A., Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., Rezhimy s obostreniem v zadachakh dlya kvazilineinykh parabolicheskikh uravnenii, Nauka, M., 1987

[3] Galaktionov V. A., Posashkov S. A., “O novykh tochnykh resheniyakh parabolicheskikh uravnenii s kvadratichnymi nelineinostyami”, ZhVMiMF, 29:4 (1989), 497–506 | MR

[4] Galaktionov V. A., “On new exact blow-up solutions for nonlinear heat conduction equations with source and applications”, Differential Integral Equations, 3:5 (1990), 863–874 | MR | Zbl

[5] Galaktionov V. A., Invariant subspaces and new explicit solution to evolution equations with quadratic nonlinearities, School of Mathematics. Report No. AM-91-11, Univ. Bristol, Bristol, 1991

[6] Rudykh G. A., Semenov E. I., “Postroenie tochnykh reshenii mnogomernogo kvazilineinogo uravneniya teploprovodnosti”, ZhVMiMF, 33:8 (1993), 1228–1239 | MR | Zbl

[7] Pukhnachev V. V., “Mnogomernye tochnye resheniya uravneniya nelineinoi diffuzii”, Prikl. mekh. i tekh. fizika, 36:2 (1995), 23–31 | MR | Zbl

[8] Kosygina E. R., “Ob anizotropnykh tochnykh resheniyakh mnogomernogo uravneniya nestatsionarnoi filtratsii”, ZhVMiMF, 35:2 (1995), 241–259 | MR | Zbl

[9] Rudykh G. A., Semenov E. I., “Novye tochnye resheniya odnomernogo uravneniya nelineinoi diffuzii”, Sib. matem. zh., 38:5 (1997), 1130–1139 | MR | Zbl

[10] Rudykh G. A., Semenov E. I., “O novykh tochnykh resheniyakh odnomernogo uravneniya nelineinoi diffuzii s istochnikom (stokom)”, ZhVMiMF (to appear)

[11] Rudykh G. A., Semenov E. I., “Tochnye neotritsatelnye resheniya mnogomernogo uravneniya nelineinoi diffuzii”, Sib. matem. zh., 1998, no. 5, 1131–1141 | MR

[12] Akhiezer N. I., Elementy teorii ellipticheskikh funktsii, Gostekhizdat, M., 1948

[13] Ovsyannikov L. V., Gruppovoi analiz differentsialnykh uravnenii, Nauka, M., 1978

[14] Ibragimov N. Kh., Gruppy preobrazovanii v matematicheskoi fizike, Nauka, M., 1983