关于一类带截断空间边界条件的线性抛物型PDE适定性的文献咨询
Hey there, great question—you’re totally right that some textbooks gloss over these details, especially when dealing with truncated domains. Let me break down some solid references that explicitly cover the well-posedness (existence, uniqueness, and often regularity) for exactly this type of linear parabolic PDE with Dirichlet boundary conditions on a bounded interval:
Evans' Partial Differential Equations: This is a standard graduate-level text, and Chapter 7 (on parabolic equations) dedicates sections to linear equations with Dirichlet BCs on bounded domains. It includes rigorous proofs of existence and uniqueness using semigroup methods and variational formulations, specifically addressing cases with lower-order terms (like your $\varepsilon u \partial_x v$ term, assuming $u$ is a bounded coefficient here). The proofs explicitly handle bounded intervals $[a,b]$ so it’s directly applicable to your truncated space setup.
Friedman's Partial Differential Equations of Parabolic Type: This is a classic monograph focused entirely on parabolic PDEs. It has detailed sections on linear parabolic equations with variable coefficients (including first-order convective terms) on bounded domains. Friedman provides constructive proofs (using difference schemes and energy methods) for existence and uniqueness of solutions, and explicitly treats Dirichlet boundary conditions like your $v(s,a)=h_a(s)$ and $v(s,b)=h_b(s)$.
Ladyzhenskaya, Solonnikov, and Ural'tseva's Linear and Quasilinear Equations of Parabolic Type: If you want ultra-rigorous, comprehensive coverage, this is the go-to reference. It covers linear parabolic equations with general lower-order terms on bounded domains, with detailed analysis of well-posedness in various function spaces (like $L^p$ and Hölder spaces). The authors explicitly handle Dirichlet boundary conditions on intervals and prove existence, uniqueness, and regularity results that directly apply to your equation.
For a more applied take: If you’re looking for references that connect well-posedness to practical computations, Numerical Solution of Partial Differential Equations: An Introduction by Morton and Mayers includes sections on well-posedness of linear parabolic equations on bounded domains, with justifications that tie into finite difference methods. While it’s more numerical, it still explicitly addresses existence and uniqueness as foundational to valid computations.
A quick note: Make sure to check the regularity assumptions on your coefficients ($u$), source term $f$, initial data $g$, and boundary data $h_a, h_b$—all these references will specify the conditions needed (e.g., Hölder continuity, boundedness) for well-posedness to hold.
备注：内容来源于stack exchange，提问作者numbers and me
