Local and global solvability of the quasilinear hyperbolic Stefan problem on the line

Well-posedness of boundary-value problems for multidimensional hyperbolic systems

“…(4), (5), (10), (11), (14), and (15) and the assumption of the smoothness of the functions λ ij are satisfied, the functions f i are continuous in all arguments, continuously differentiable with respect to x, y, and u, and satisfying the global Lipschitz condition with respect to u n ∈R uniformly in ( , , ) x y t ∈ M for every compact set M ∈Π , and the functions ϕ i and μ i are continuously differentiable. Then problem (1), (2), (12) has a unique classical solution in the domain Π .…”

“…where 3 , and A 4 = a ij i j I 4 . Consider the problem for system (1) with initial conditions (2) and the boundary conditions…”

Quasilinear hyperbolic stefan problem with nonlocal boundary conditions

“…According to the statement proved above, there exists a unique generalized solution of this problem in the space M D M.T 0 ; U 0 ; L x ; L t /; where T 0 ; U 0 ; L x ; and L t are certain fixed values of parameters. Note that, according to relation (21), the values L x and L t can be taken sufficiently large for the functions u i ; uĩ ; i 2 f1; : : : ; ng; to satisfy condition (iii) in the definition of the space M : According to relation (25), the value T 0 can be chosen so small that the functions s j and sj ; j 2 f1; 2g; satisfy condition (i) in the definition of the space M and the functions u i and uĩ ; i 2 f1; : : : ; ng; satisfy condition (ii) for this space. Thus, both generalized solutions of the considered problem belong to the space M ; a contradiction.…”

“…The correct solvability of this problem for small values of time is established by using the method proposed in [25][26][27].…”