Stability of solutions of quasilinear parabolic equations

We bound the difference between solutions u and v of ut=aΔu+divxf+h and vt=bΔv+divxg+k with initial data φ and ψ, respectively, by

     

Global solutions of abstract quasilinear parabolic equations

Local and global existence and uniqueness for strict solutions of abstract quasilinear parabolic equations are studied. Applications to quasilinear parabolic partial differential equations are also given.     

Global existence and blowup solutions for quasilinear parabolic equations

The authors discuss the quasilinear parabolic equation ut=∇⋅(g(u)∇u)+h(u,∇u)+f(u) with u_|∂Ω=0, u(x,0)=?(x). If f, g and h are polynomials with proper degrees and proper coefficients, they show that the blowup property only depends on the first eigenvalue of −Δ in Ω with Dirichlet boundary condition. For a special case, they obtain a sharp result.     

Transformation of approximate solutions of linear parabolic equations into asymptotic solutions of quasilinear parabolic equations

Moscow Institute of Electronics and Mathematics. Translated from Matematicheskie Zametki, Vol. 56, No. 6, pp. 122–126, December, 1994.     

Blow-up of sign-changing solutions to quasilinear parabolic equations

We study the blow-up of sign-changing solutions to the Cauchy problem for quasilinear parabolic equations of arbitrary order. Our approach is based on H. Levine’s remarkable idea of constructing a concavity inequality for a negative power of a standard positive definite functional. Combining this with the nonlinear capacity method, which is based on the choice of optimal test functions, we find conditions for the blow-up of solutions to the problems under consideration.     

Stabilization of solutions of initial-boundary problems for quasilinear parabolic equations

Lower bounds are obtained for solutions of the initial-boundary Dirichlet problem for high order equations. Sharp bounds are also obtained for ess sup¦u(x, t)¦ of the Neumann initial-boundary problem for a second- order equation in D=x(t >0), where (Rⁿ, n 2 is a domain with noncompact convex boundary.Translated from Ukrainskii Maternaticheskii Zhurnal, Vol. 44, No. 10, pp. 1441–1450, October, 1992.     

Continuous branches of periodic solutions of quasilinear parabolic equations

Siberian Mathematical Journal -     

Some results on the qualitative properties of positive solutions of quasilinear elliptic equations

We consider the Dirichlet problem in Ω with zero Dirichlet boundary conditions. We prove local summability properties of and we exploit these results to give geometric characterizations of the critical set . We extend to the case of changing sign nonlinearities some results known in the case f(s) > 0 for s > 0. Berardino Sciunzi: Supported by MURST, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”