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 -