# EXISTENCE IN SCHWARTZ SPACE AND SOLUTIONS PROPERTIES OF THE HOPF–TYPE EQUATION WITH VARIABLE COEFFICIENTS

### Abstract

The issue of the existence of solutions of the Cauchy problem for a first-order quasi-linear differential equation with partial derivatives and variable coefficients is considered. The studied equation is a generalization of the classic Hopf equation, which is used in the study of many mathematical models of hydrodynamics. This equation arises when constructing approximate (asymptotic) solutions of the Korteweg–de Vries equation and other equations with variable coefficients and a singular perturbation, in particular, when finding their asymptotic step-type soliton-like solutions. For the mentioned differential equation of the first order, the solution of the Cauchy problem is obtained in analytical form, and the statement about sufficient conditions for the existence of solutions of the initial problem in the space of rapidly decreasing functions is proved. A similar problem for a first-order differential equation with partial derivatives with variable coefficients and quadratic nonlinearity is considered.

### References

Sedov L. I. Mechanics of continuous media. In 2 volumes. Series in Theoretical and Applied Mechanics. Vol. 4. World Scientific, 1997. 1368 p.

Hopf E. The partial differential equation $u_t+uu_x=μu_xx$. Comm. Pure Appl. Math.1950. Vol. 3. P. 201–230.

Burgers J. M. Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion. Verhandelingen der Koninklijke Nederlandsche Akademie van Wetenschappen. Afdeeling natuurkunde (Eerste sectie).1939. Deel XVII, no. 2. P. 1–53.

Burgers J. M. Application of a model system to illustrate some points of the statistical theory of free turbulence. Proceedings of the section of sciences. Koninklijke Nederlandsche Akademie van Wetenschappen. Amsterdam. 1940. Vol. XLII, no. 1. P. 2–12.

Oleinik O. A. Discontinuous solutions of non-linear differential equations. Uspekhi Mat. nauk. 1957. Volume 12, No 2. P. 3–73.

Hopf E. On the right weak solution of the Cauchy problem for a quasilinear equation of first order. Journ. Math. Mech.1969. Vol. 19, No 6. P. 483–487.

Goritsky A. Yu., Kruzhkov S. N., Chechkin G. A. Partial differential equations of the first order. M.: Publishing house of the Center appl. researches at mech.-mat. faculty of Moscow State University, 1999. 96 p.

Scott E. Waves in active and nonlinear media in applications to electronics. Moscow: Soviet radio. 1977. 368 p.

Lighthill M. J., Whitham G. B. On kinematic waves. I. Flood movement in long rivers. Proc. Roy. Soc.(London). 1965. Vol. 229 A. P. 281–316.

Lighthill M. J. Group velocity. Journ. of the Institute of Mathematical Applications.1965. Vol. 1. P. 1–28.

Gazis D. C. Mathematical theory of automobile traffic. Science.1967. Vol. 157. P. 273–281.

Lighthill M. J., Whitham G. B. On kinematic waves. II. A theory of traffic flow on long crowded roads. 1965. Vol. 229 A. P. 317–345.

Golovatyy Y. D., Kyrylych V. M., Lavrenyuk S. P. Differential equations. Lviv: Ivan Franko National University, 2011. 470 p.

Zaslavsky G. M., Sagdeev R. Z. Introduction to nonlinear physics. From pendulum to turbulence and chaos. Moscow: Nauka, 1988. 368 p.

Whitham J. Linear and non-linear waves. M.: Mir. 1977. 624 p.

Maslov V. P., Omel’yanov G. A. Geometric asymptotics for PDE. American Math. Society, Providence. 2001. 243 p.

Solitons. Ed. R. Bullaf, F. Codry. Moscow: Mir, 1983. 408 p.

Samoilenko V. G., Samoilenko Yu. I. Asymptotics for single-phase soliton-like solutions of the Korteweg–de Vries equation with variable coefficients.. Ukr. mat. journal. 2005. Volume 58, No 1. P. 111–124.

Samoilenko Yu. I. Asymptotic solutions of the singularly perturbed Korteweg-de Vries equation with variable coefficients (general case). Mathematical Bulletin of the NTSh. 2010. Vol. 7. P. 227–242.

Faminskii A. V. The Cauchy problem for the Korteweg–de Vries equation and its generalizations. Proceedings of the seminar. I. G. Petrovsky. 1988. Volume 13, pp. 56–105.

Samoilenko A. M., Perestyuk M. O., Parasyuk I. O. Differential Equations. VPC <>. 2010. 527 p.

Samoilenko V. G., Samoilenko Yu. I. Asymptotic multiphase soliton-like solutions of the Cauchy problem for the singularly perturbed Korteweg–de Vries equation with variable coefficients. Ukr. mat. journal. 2014. Vol. 66, No 12. P. 1640–1657.

Lyashko S. I., Samoilenko V. Hr., Samoilenko Yu. I., Lyashko N. I. Asymptoticanalysis of the Korteweg-de Vries equation by the nonlinear WKB technique. Mathematical Modeling and Computing. 2021. Vol. 8, No 3. P. 368–378.