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Existence and Uniqueness of Strong Solutions for a Class of Quasi‐Linear Hyperbolic Equations with Order Degeneration
Author(s) -
Semerdjieva Rossitza I.
Publication year - 2002
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200204)237:1<89::aid-mana89>3.0.co;2-r
Subject(s) - mathematics , uniqueness , class (philosophy) , order (exchange) , degeneration (medical) , mathematical analysis , pure mathematics , medicine , finance , economics , pathology , artificial intelligence , computer science
Let k ( y ) > 0, ( y ) > 0 for y > 0, k (0) = (0) = 0 and lim y → 0 k ( y )/( y ) exists; then the equation L ( u ) ≔ k ( y ) u xx – ∂ y (( y ) u y ) + a ( x , y ) u x = f ( x , y , u ) is strictly hyperbolic for y > 0 and its order degenerates on the line y = 0. Consider the boundary value problem Lu = f ( x , y , u ) in G , u | AC = 0, where G is a simply connected domain in ℝ 2 with piecewise smooth boundary ∂ G = AB ∪ AC ∪ BC ; AB = {( x , 0) : 0 ≤ x ≤ 1}, AC : x = F ( y ) = ∫ y 0 ( k ( t )/( t )) 1/2 dt and BC : x = 1 – F ( y ) are characteristic curves. Existence of generalized solution is obtained by a finite element method, provided f ( x , y , u ) satisfies Carathéodory condition and | f ( x , y , u )| ≤ Q ( x , y ) + b | u | with Q ∈ L 2 ( G ), b = const > 0. It is shown also that each generalized solution is a strong solution, and that fact is used to prove uniqueness under the additional assumption | f ( x , y , u 1 ) – f ( x , y , u 2 | ≤ C | u 1 – u 2 |, where C = const > 0.