Research output: Contribution to journal › Article › peer-review
Solvability to some strongly degenerate parabolic problems. / Lavrentiev, Mikhail M.; Tani, Atusi.
In: Journal of Mathematical Analysis and Applications, Vol. 475, No. 1, 01.07.2019, p. 576-594.Research output: Contribution to journal › Article › peer-review
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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 -
ID: 18678252