Well-Posedness of Parabolic Difference Equations

1 The Abstract Cauchy Problem.- 1. Well-Posedness of the Differential Cauchy Problem in C(E).- 1. The Cauchy problem in a Banach space E. Definition of well-posedness in C(E).- 2. Examples of well-posed and ill-posed problems in C(E).- 3. The homogeneous equation. Strongly continuous semigroups.- 4. The nonhomogeneous equation. Analytic semigroups.- 5. Well-posedness in C(E) of the general Cauchy problem.- 2. Well-Posedness of the Cauchy Problem inC0?(E).- 1. The homogeneous problem. The space C0?(E).- 2. Well-posedness in C0?(E) of the general Cauchy problem.- 3. Well-Posedness of the Cauchy Problem in Lp(E).- 1. Definition of the well-posedness of the Cauchy problem in LP(E).- 2. A formula for the solution of the Cauchy problem in Lp(E).- 3. Spaces of initial data.- 4. The values of the solution of the Cauchy problem in Lp(E) for fixed t.- 5. The coercivity inequality for the solutions in Lp(E) of the general problem (1.1).- 4. Well-Posedness of the Cauchy Problem in Lp(E?,Q).- 5. Well-Posedness of the Cauchy Problem in Spaces of Smooth Functions.- 1. The space C0ß,?(E). The nonhomogeneous problem.- 2. Well-posedness of the general problem.- 3. Semigroup estimates.- 4. The coercivity inequality for the general problem.- 2 The Rothe Difference Scheme.- 0. Stability of the Difference Problem.- 1. The difference problem.- 2. Banach spaces of grid functions.- 3. The operator equation in ?(E). Definition of the stability of the difference scheme.- 4. Stability of the difference scheme.- 1. Well-Posedness of the Difference Problem in C(E).- 1. The homogeneous difference problem.- 2. The nonhomogeneous problem. A real-field criterion for analyticity.- 3. An almost coercive inequality in C(E).- 2. Well-Posedness of the Difference Problem in C0?(E).- 3. Well-Posedness of the Difference Problem in Lp(E).- 1. Definition of the well-posedness of the difference problem in LP(E).- 2. Spaces of initial data.- 3. The coercivity inequality for the solutions in LP(E) of the general problem (0.6).- 4. Well-Posedness of the Difference Problem in Lp(E?,Q).- 1. Strongly positive operators and fractional spaces.- 2. Well-posedness of the difference problem in Lp(E’?,q).- 5. Well-Posedness of the Difference Problem in Difference Analogues of Spaces of Smooth Functions.- 1. The space CQ’(E). The nonhomogeneous difference problem.- 2. Well-posedness of the general difference problem.- 3. Estimates for powers of the resolvent.- 4. The coercivity inequality for the general problem.- 3 PadÉ Difference Schemes.- 0. Stability of the Difference Problem.- 1. Padé approximants of the function e-z.- 2. Difference schemes of Padé class.- 1. Well-Posedness of the Difference Problem in C(E).- 1. The homogeneous problem.- 2. The nonhomogeneous problem.- 3. Sufficient conditions for almost-well-posedness. A real-field criterion for analyticity.- 4. Estimates of powers of the operator step.- 2. Well-Posedness of the Difference Problem in C0?(E).- 1. The case of a general space C0?(E).- 2. The case of the special space C0? (E).- 3. Well-Posedness of the Difference Problem in Lp(E).- 1. Definition of the well-posedness of the difference problem in Lp(E). Stability of the difference problem.- 2. Spaces of initial data. Well-posedness of the difference problem.- 3. Estimates of powers of the operator step.- 4. Well-Posedness of the Difference Problem in Lp(E’?,Q).- 1. Stability of the difference problem.- 2. Well-posedness of the difference problem.- 5. Well-Posedness of the Difference Problem in Difference Analogues of Spaces of Smooth Functions.- 1. Well-posedness of the difference problem in C0ß,? (E).- 2. Estimates of powers of the operator step. The coercivity inequality for the general problem.- 4 Difference Schemes for Parabolic Equations.- 1. Elliptic Difference Operators with Constant Coefficients.- 1. The definition of an elliptic difference operator and properties of its symbol.- 2. A formula for the solution of the resolvent equation.- 3. Point estimates for the fundamental solution of the resolvent equation.- 4. Sharpening of the point estimates of the fundamental solution of the resolvent equation.- 5. Positivity of homogeneous elliptic difference operators with constant coefficients.- 6. Point estimates of the fundamental solution of the resolvent equation in the case m ? n.- 7. Point estimates of difference derivatives of the fundamental solution of the resolvent equation.- 2. Fractional Spaces in the case of an Elliptic Difference Operator.- 1. The fractional spaces E’?,?(Ch, Ah).- 2. Positivity of the elliptic difference operator in L1h-. The fractional spaces E’?,1(L1h, Ah).- 3. Positivity of elliptic difference operators in Lph-. The fractional spaces E’a,p(Lph, Ah).- 4. The coercivity inequality for an elliptic difference operator in Cma(Rhn) and Wpm?(Rhn).- 5. Elliptic difference operators in L2h.- 3. Stability and Coercivity Estimates.- I. Approximation with respect to the space variables.- II. Approximation with respect to the time variable.- Comments on the Literature.- References.