Sentence examples for initial data condition from inspiring English sources

It follows from the initial data condition u 0 ( x ) ∈ W 1, ∞ that u 0 ( x ) is Lipschitz continuous, i.e., there exists L 0 > 0 such that | u 0 ( x ) − u 0 ( y ) | ≤ L 0 | x − y |, ( x, y ) ∈ Ω ¯ × Ω ¯. (3.8).

For any given initial data satisfying condition (3), there exists a unique solution of model (2) defined on ([0,+infty)), and this solution remains nonnegative and bounded for all (tgeq0).

For any given initial data satisfying condition (6), there exists a unique solution of problem (3)–(7) defined on ([0,+infty )), and this solution remains nonnegative and bounded for all (t>0).

Numerical methods to approximate these nonlinear hyperbolic PDEs (or for that matter any PDE) require inputs such as the initial data, boundary conditions and coefficients in the fluxes, sources and friction terms of the PDE.

Let conditions be held and initial data satisfy the condition, then the problem (3.1) has a unique solution, where (3.34).

note that the compatibility conditions between the initial data and boundary conditions hold.

Right: Time evolution of the firing rate; we observe numerically a blow-up behaviour for an initial data satisfying the condition of concentration around V F described in Theorem 2.2.

They obtained a local existence result and proved, for certain initial data and suitable conditions on g and γ (under weaker conditions than those in [2, 3]), that the solution is global and decays uniformly (exponentially or polynomially depending on the decay rate of the relaxation function g) if the initial data is small enough.

and the consistencies between the initial data and boundary conditions hold.

The existence and uniqueness of global weak solutions for Eq. (3) have been established provided that the initial data satisfy appropriate conditions in [6].

We finally prove the existence and uniqueness of global weak solutions to the equation provided the initial data satisfies appropriate conditions.