Annali di Matematica Pura ed Applicata

, Volume 88, Issue 1, pp 193–205

# A two phase Stefan problem with flux boundary conditions

• John R. Cannon
• Mario Primicerio
Article

## Abstract

We studied a two phase Stefan problem in a infinite plane slab, when the thermal fluxes are assigned on the two limiting planes.

We proved existence and uniqueness of the solution upon minimal smoothness assumptions upon the initial and boundary data, and we demonstrated the continuous and monotone dependence of the solution on the data.

In sec. 5 we studied in which cases one of the two phases disappears and the asymptotic behavior in the cases in which the two phases exist for all time.

### Keywords

Boundary Condition Asymptotic Behavior Boundary Data Thermal Flux Stefan Problem

## Riassunto

Si studia un problema di Stefan a due fasi in uno strato piano indefinito quando si suppongono assegnati i flussi termici sui piani che delimitano lo strato stesso.

Viene dimostrata l’esistenza e l’unicità della soluzione con ipotesi assai generali sui dati iniziali ed al contorno del problema, nonchè la dipendenza continua e monotona della soluzione da tali dati.

Si esaminano infine i casi in cui una delle due fasi può sparire ed il comportamento asintotico in caso di permanenza delle due fasi.

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### References

1. [1]
B. M. Budak andM. Z. Moskal,Classical solution of the multidimensional multifront Stefan problem, Soviet Math. Dokl., Vol. 10 (1969), #5, pp. 1043–1046.Google Scholar
2. [2]
J. R. Cannon,A priori estimate for continuation of the solution of the heat equation in the space variable, Ann. Mat. Pura Appl. 65 (1964), pp. 377–388.
3. [3]
J. R. Cannon andJim Douglas Jr.,The stability of the boundary in a Stefan problem, Ann. della Scuola Normale Superiore di Pisa, Vol. XXI, Fasc. I (1967), pp. 83–91.
4. [4]
J. R. Cannon andC. D. Hill,Existence, uniqueness, stability, and monotone dependence in a Stefan problem for the heat equation, J. of Math. and Mech. Vol. 17 (1967), pp. 1–20.
5. [5]
J. R. Cannon, Jim Douglas Jr. andC. D. Hill,A multi-boundary Stefan problem and the disappearance of phases, J. of Math. and Mech. Vol. 17, (1967), pp. 21–34.
6. [6]
J. R. Cannon andC. D. Hill,Remarks on a Stefan problem, J. of Math. and Mech., Vol. 17 (1967), pp. 433–442.
7. [7]
J. R. Cannon andC. D. Hill,On the infinite differentiability of the free boundary in a Stefan problem, J. of Math. Anal. and Appl., Vol. 22, (1968), pp. 385–397.
8. [8]
J. R. Cannon andMario Primicerio,A two phase Stefan problem with temperature boundary conditions, Ann. Mat. Pura Appl. (IV), vol. 88 (1971), pp. 177–192.
9. [9]
Avner Friedman,The Stefan problem in several space variables, Transactions of the A.M.S., Vol. 133, (1968), pp. 51–87.
10. [10]
—— ——,One dimensional Stefan problems with nonmonotone free boundary, Transactions of the A.M.S., Vol. 133, (1968), pp. 89–114.
11. [11]
—— ——,Correction to « the Stefan problem in several space variables », Transactions of the A.M.S., Vol. 142, (1969), p. 557.
12. [12]
M. Gevrey,Sur les équations aux dérivées partielles du type parabolique, J. Math. (ser. 6), 9 (1913), pp. 305–471.Google Scholar
13. [13]
S. L. Kamenomostskaja,On Stefan’s problem, Mat. Sb. 53 (95) (1965), pp. 485–514.
14. [14]
Jiang Li-shang,Existence and differentiability of the sulution of a two-phase Stefan problem for quasi-linear parabolic equations, Chinese Math. 7 (1965), pp. 481–496.Google Scholar
15. [15]
D. Quilghini,Una analisi fisico-matematica del processo del cambiamento di fase, Ann. di Mat. pura ed applicata, (IV), Vol. LXVII (1965), pp. 33–74.
16. [16]
L. I. Rubinstein,Two phase Stefan problem on a segment with one-phase initial state of thermoconductive medium, Učen, Zap. Lat. Gos. Univ. Stučki 58 (1964), pp. 111–148.
17. [17]
G. Sestini,Esistenza ed unicità nel problema di Stefan relativo a campi dotati di simmetria, Rivista Mat. Univ. Parma 3 (1952), pp. 103–113.