TY - JOUR

T1 - Solvability to some strongly degenerate parabolic problems

AU - Lavrentiev, Mikhail M.

AU - Tani, Atusi

PY - 2019/7/1

Y1 - 2019/7/1

N2 - Nonlinear parabolic equations of “divergence form,” u t =(φ(u)ψ(u x )) x , are considered under the assumption that the “material flux,” φ(u)ψ(v), is bounded for all values of arguments, u and v. In literature such equations have been referred to as “strongly degenerate” equations. This is due to the fact that the coefficient, φ(u)ψ ′ (u x ), of the second derivative, u xx , can be arbitrarily small for large value of the gradient, u x . The “hyperbolic phenomena” (unbounded growth of space derivatives within a finite time) have been established in literature for solutions to Cauchy problem for the above-mentioned equations. Accordingly one can expect a correct statement of the initial-boundary value problem for such equations only under additional assumptions on the problem data. In this paper we describe several restrictions, under which the initial-boundary value problems for strongly degenerate parabolic equations are well-posed.

AB - Nonlinear parabolic equations of “divergence form,” u t =(φ(u)ψ(u x )) x , are considered under the assumption that the “material flux,” φ(u)ψ(v), is bounded for all values of arguments, u and v. In literature such equations have been referred to as “strongly degenerate” equations. This is due to the fact that the coefficient, φ(u)ψ ′ (u x ), of the second derivative, u xx , can be arbitrarily small for large value of the gradient, u x . The “hyperbolic phenomena” (unbounded growth of space derivatives within a finite time) have been established in literature for solutions to Cauchy problem for the above-mentioned equations. Accordingly one can expect a correct statement of the initial-boundary value problem for such equations only under additional assumptions on the problem data. In this paper we describe several restrictions, under which the initial-boundary value problems for strongly degenerate parabolic equations are well-posed.

KW - Generalized distance

KW - Global-in-time solutions

KW - Hyperbolic phenomena

KW - Initial-boundary value problem

KW - Strongly degenerate parabolic equations

KW - HEAT

KW - BOUNDARY

KW - BLOW-UP

UR - http://www.scopus.com/inward/record.url?scp=85062230851&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2019.02.056

DO - 10.1016/j.jmaa.2019.02.056

M3 - Article

AN - SCOPUS:85062230851

VL - 475

SP - 576

EP - 594

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -