TY - JOUR
T1 - Existence and comparison theorems for nonlinear diffusion systems
AU - Kuiper, Hendrik J.
PY - 1977/8
Y1 - 1977/8
N2 - In this paper we prove existence, uniqueness, and regularity results for systems of nonlinear second order parabolic equations with boundary conditions of the Dirichlet, Neumann, and regular oblique derivative types. Let K(t) consist of all functions (v1(x), v2(x),..., vm(x)) from Ω ⊂ Rn into Rm which satisfy ψi(x, t) ≤ vi(x) ≤ θi(x, t) for all x ε{lunate} Ω and 1 ≤ i ≤ m, where ψiand θi are extended real-valued functions on \ ̄gW × [0, T). We find conditions which will ensure that a solution U(x, t) ≡ (u1(x, t), u2(x, t),..., um(x, t)) which satisfies U(x, 0) ε{lunate} K(0) will also satisfy U(x, t) ε{lunate} K(t) for all 0 ≤ t < T. This result, which has some similarity to the Gronwall Inequality, is then used to prove a global existence theorem.
AB - In this paper we prove existence, uniqueness, and regularity results for systems of nonlinear second order parabolic equations with boundary conditions of the Dirichlet, Neumann, and regular oblique derivative types. Let K(t) consist of all functions (v1(x), v2(x),..., vm(x)) from Ω ⊂ Rn into Rm which satisfy ψi(x, t) ≤ vi(x) ≤ θi(x, t) for all x ε{lunate} Ω and 1 ≤ i ≤ m, where ψiand θi are extended real-valued functions on \ ̄gW × [0, T). We find conditions which will ensure that a solution U(x, t) ≡ (u1(x, t), u2(x, t),..., um(x, t)) which satisfies U(x, 0) ε{lunate} K(0) will also satisfy U(x, t) ε{lunate} K(t) for all 0 ≤ t < T. This result, which has some similarity to the Gronwall Inequality, is then used to prove a global existence theorem.
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U2 - 10.1016/0022-247X(77)90057-9
DO - 10.1016/0022-247X(77)90057-9
M3 - Article
AN - SCOPUS:49449122533
SN - 0022-247X
VL - 60
SP - 166
EP - 181
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 1
ER -