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Annali di Matematica Pura ed Applicata

, Volume 87, Issue 1, pp 375–388 | Cite as

On the existence of a generalized solution to the first initial-boundary value problem for a nonlinear parabolic equation

  • V. Šeda
Article
  • 16 Downloads

Summary

In the paper first the existence of a classical solution to an initial-boundary value problem for the nonlinear parabolic equation\(\frac{{\partial z}}{{\partial t}} = \frac{{\partial ^2 z}}{{\partial x^2 }} + f\left( {x,t,z} \right)\) is proved under the standard condition on Hölder continuity of f but a quite general condition on the growth of f. Then, by using the possibility of the approximation of a continuous function by means of Hölder continuous functions, the foregoing result is applied to the proof of the existence of a generalized solution to the first boundary value problem for the same equation where only continuity of f and a weak assumption on the growth of f is required.

Keywords

Continuous Function General Condition Standard Condition Generalize Solution Parabolic Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    C. Ciliberto,Su di un problema al contorno per una equazione non lineare di tipo parabolico in due variabili, Ricerche Mat. 1 (1952), pp. 55–77.MATHMathSciNetGoogle Scholar
  2. [2]
    —— ——,Su di un problem i al contorno per l'equazione u xx -u y =f(x, y, u, u x), Ricerche Mat. 1 (1952), pp. 295–316.MATHMathSciNetGoogle Scholar
  3. [3]
    A. Friedman,Partial differential equations of parabolic type, Prentice Hall 1964.Google Scholar
  4. [4]
    -- --,Partial differential equations, Holt, Rinehart and Winston, Inc. 1969.Google Scholar
  5. [5]
    E. Gagliardo,Problema al contorno per equazioni differenziali lineari di tipo parabolico in n variabili, Ricerche Mat. 5 (1956), pp. 169–205.MATHMathSciNetGoogle Scholar
  6. [6]
    —— ——,Teoremi di esistenza e di unicità per problemi al contorno relativi ad equazioni paraboliche lineari e quasi lineari in n variabili, Ricerche Mat. 5 (1956) pp. 239–257.MATHMathSciNetGoogle Scholar
  7. [7]
    A.M. Iľjin, A. S. Kalasňikov, O. A. Olejnik,Linear equations of the second order of parabolic type (In Russian), Usp. Mat. Nauk 17 (1962), pp. 3–146.Google Scholar
  8. [8]
    E. Kamke,Differentialgleichungen reeler Funktionen, Leipzig 1956.Google Scholar
  9. [9]
    W. Mlak,The first boundary value problem for a nonlinear parabolic equation, Ann. Pol. Math. 5(1958), pp. 257–272.MATHMathSciNetGoogle Scholar
  10. [10]
    O. A. Olejnik, S. N. Kružkov,Quasi-linear parabolic equations of the second order with many independent variables (In Russian), Usp. Mat. Nauk. 16(1961), pp. 115–155.Google Scholar
  11. [11]
    V. Šeda,A remark to the theory of the heat equation, Ann. Mat. Pura Appl. 86 (1970), pp. 357–366.MathSciNetGoogle Scholar

Copyright information

© Nicola Zanichelli Editore 1970

Authors and Affiliations

  • V. Šeda
    • 1
  1. 1.BratislavaCzechoslovakia

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