Degenerate Parabolic Equations

Frontmatter

I. Notation and function spaces

Abstract

Let Ω be a bounded domain in R ^N of boundary ∂Ω and for \( 0<T<\infty \) let Ω _T denote the cylindrical domain \( \Omega \times (0,T] \). Also let,

$$ {S_T} \equiv \partial \Omega \times [0,T],\Gamma \equiv {S_T} \cup (\Omega \times \left\{ 0 \right\}) $$

denote the lateral boundary and the parabolic boundary of Ω _T respectively.

Emmanuele DiBenedetto

II. Weak solutions and local energy estimates

Abstract

We introduce a class of quasilinear parabolic equations with the principal part in divergence form, that are either degenerate or singular due to the vanishing of the gradient |Du| of their solutions.

Emmanuele DiBenedetto

III. Hölder continuity of solutions of degenerate parabolic equations

Abstract

Consider solutions u of (1.1) or of Chap. II for the casep >2. The equation is degenerate since the modulus of ellipticity vanishes when |Du| = 0. We will prove that if\( u \in L_{loc}^\infty ({\Omega _T}) \)then it is Hölder continuous within its domain of definition. It will shown in Chap.V that local weak solutions of such degenerate equations are indeed locally bounded. To simplify the presentation we will assume that\( u \in {L^\infty }({\Omega _T}) \). If u is only locally bounded, it will suffice to work within a fixed compact subset ofΩ _T. In the theorems below, the statement that a constant γ depends upon the data means that it can be determined a priori only in terms of the norm \( {\left\| u \right\|_{\infty,{\Omega _T}}} \)the constants C_ii=0, 1, 2, and the norms \( \left\| {{\varphi _0},\varphi _1^{\frac{p}{{p - 1}}},{\varphi _2}} \right\|\mathop q\limits^ \wedge,\mathop r\limits^ \wedge \);Ω _Tappearing in the structure conditions (A₁)—(A₃). We let K denote a compact subset ofΩ _Tand letp— dist(\( \mathcal{K} \);\( \Gamma \)) be theintrinsicparabolic distance from to the parabolic boundary ofΩ _Ti.e.,

$$ p - dist(\mathcal{K};\Gamma ;p) \equiv \mathop {\mathop {\inf }\limits_{(x,t) \in \mathcal{K}} }\limits_{(y,s) \in \Gamma } (\left| {x - y} \right| + \left\| u \right\|_{\infty,{\Omega _T}}^{\frac{{p - 2}}{p}}{\left| {t - s} \right|^{1/p}}). $$

(1.1)

Emmanuele DiBenedetto

IV. Hölder continuity of solutions of singular parabolic equations

Abstract

Evolution equations of the type of (1.1) of Chap. II for 1 < p< 2 are singular since their modulus of ellipticity becomes unbounded when \( |Du| = 0\). We will lay out a theory of local and global Hölder continuity of solutions u of such singular p.d.e.’s. We assume that \( u \in {L^\infty }({\Omega _T})\). If u is only locally bounded it will suffice to work within compact subsets K of Ω _T. The intrinsic p—distance dist (K;Γp) from /C to the parabolic boundary of Ω _T is defined as in (1.1) of Chap. III. In the theorems below, the statement that a constant y depends upon the data means that it can be determined a priori only in terms of \( \parallel u{\parallel _{\infty,{\Omega _T}}}\), the constants C_i, i= 0, 1, 2, and the norms \( \parallel {\varphi _0},\varphi _1^{\frac{p}{{p - 1}}},{\varphi _2}{\parallel _{\hat q,\hat r;{\Omega _T}}}\) appearing in the structure conditions (A ₁)-(A ₅). For p in the singular range 1 < p < 2, let, p — dist (K; Γ) denote the intrinsic parabolic distance from K to the parabolic boundary of Ω _T, i.e,

$$p - dist(K;\Gamma ) \equiv \mathop {\inf }\limits_{\mathop {(x,t) \in K}\limits_{(y,s) \in \Gamma } } (\parallel u\parallel _{\infty,{\Omega _T}}^{\frac{{2 - p}}{p}}|x - y| + |t - s{|^{\frac{1}{p}}})$$

Emmanuele DiBenedetto

V. Boundedness of weak solutions

Abstract

Let u be a weak solution of equations of the type of (1.1) of Chap. II in Ω _T We will establish local and global bounds for u in. Ω _T . Global bounds depend on the data prescribed on the parabolic boundary of Ω _T . Local bounds are given in terms of local integral norms of u. Consider the cubes K _ρ ⊂ K _2ρ . After a translation we may assume they are contained in Ω provided ρ is sufficiently small. For 0 ≤ t _i < t _o <t < T consider the cylindrical domains

$$ {Q_0} \equiv {K_\rho } \times ({t_0},t),{Q_1} \equiv {K_{2\rho }} \times ({t_1},t),{Q_0} \subset {Q_1} \subset {\Omega _T}$$

.

Emmanuele DiBenedetto

VI. Harnack estimates: the case p>2

Abstract

We will establish a Harnack-type estimate for non-negative weak solutions of degenerate parabolic equations of the type

$$\left\{ {\begin{array}{*{20}{c}}{u \in {C_{loc}}(0,T;L\begin{array}{*{20}{c}}{_2} \\{^{loc}}\end{array}(\Omega )) \cap L\begin{array}{*{20}{c}}{_p}\\{^{loc}}\end{array}(0,T;W\begin{array}{*{20}{c}}{{}_{1,p}} \\{{}^{loc}}\end{array}(\Omega )),p >2,} \\{ut - div{{\left| {Du} \right|}^{p - 2}}Du = 0,in {\Omega _T}.}\end{array}} \right.$$

(1.1)

Emmanuele DiBenedetto

VII. Harnack estimates and extinction profile for singular equations

Emmanuele DiBenedetto

VIII. Degenerate and singular parabolic systems

Abstract

We turn now to quasilinear systems whose principal part becomes either degenerate or singular at points where. To present a streamlined cross section of the theory, we refer to the model system

$$ \left\{{\begin{array}{*{20}{c}} {u \equiv \left( {{u_1},{u_2}, \ldots,{u_m}} \right),m\in N,}\\ {{u_i}\in{C_{loc}}\left({0,T;L_{loc}^2\left(\Omega\right)}\right)\cap {L^p}\left({0,T;W_{loc}^{1,p}\left(\Omega \right)} \right),i=1,2,\ldots,m,}\\ {{u_t}- div{{\left| {Du}\right|}^{p - 2}}Du =0in{\Omega_{\rm T}}} \end{array}}\right. $$

(1.1)

Emmanuele DiBenedetto

IX. Parabolic p-systems: Hölder continuity of D u

Abstract

The space gradient Du of local weak solutions of the quasilinear system (1.10) of Chap. VIII are locally Holder continuous in Ω _T provided the structure conditions (Si)-(S6) are in force. We will show this first for the homogeneous system (1.1) and then will indicate how to extend it to the general systems (1.10). The estimates of this chapter hold in the interior of 12T and deteriorate near its parabolic boundary T. If K is a compact subset ofΩ _T we let dist (К; Γ) denote the parabolic distance from К to the parabolic boundaryΓof Ω _T,i.e., dist

$$ \left( {\mathcal{K};\Gamma } \right) \equiv \mathop {\mathop {\inf }\limits_{\left( {x,t} \right) \in \mathcal{K}} }\limits_{\left( {y,s} \right) \in \Gamma } \left( {\left| {x - y} \right| + \sqrt {\left| {t - s} \right|} } \right) $$

Emmanuele DiBenedetto

X. Parabolic p-systems: boundary regularity

Abstract

We will establish everywhere regularity up the boundary for weak solutions of the parabolic system

$$ \left\{ {\begin{array}{*{20}{c}} {u \equiv ({u_1},{u_2}, \ldots,{u_m}), m \in N,} \\ {{u_i} \in C(\varepsilon,T,{L^2}(\Omega )) \cap {L^p}(\varepsilon,T,{W^{1,p}}(\Omega )),} \\ {{u_{i,t}} - div{{\left| {Du} \right|}^{p - 2}}D{u_i} = {B_i}(x,t,u,Du),in \Omega \times (\varepsilon,T)} \\ {\varepsilon \in (0,T),i = 1,2, \ldots,m,p > \max \left\{ {1;\frac{{2N}}{{N + 2}}} \right\},} \end{array}} \right. $$

(1.1)

associated with Dirichlet boundary data

$$ {u_i}( \cdot,t) = {g_i}( \cdot,t) on \partial \Omega \times(\varepsilon,T), $$

(1.2)

in the sense of the traces on, of functions in W ¹ P(,f2). The basic assumptions on 8.f2, the boundary data g and the forcing term B

$$ g \equiv ({g_1},{g_2}, \ldots,{g_m}), B \equiv ({B_1},{B_2}, \ldots,{B_m}), $$

are the following: ∂Ωis of classC1,λ;for some λ ∈(0,1), in the sense of (1.2) of Chap. I. Thus the norm

$$ {\left| {\left\| {\partial \Omega } \right\|} \right|_{1 + \lambda }} $$

is finite.

$${g_i},i = 1,2, \ldots,m,$$

are restrictions to ∂Ω of functions ğ _i defined in the whole Ω _T , and satisfying is of class for some in the sense of (1.2) of Chap. I. Thus the norm is finite.

The functions, are restrictions to of functions defined in the whole, and satisfying

Emmanuele DiBenedetto

XI. Non-negative solutions in Σ T . The case p>2

Abstract

Non-negative solutions of the heat equation in a strip ET – R^N x (0, T) are somewhat special in the sense that they grow no faster than \( {e^a}\left| x \right|2,a < 1/4T,as\left| x \right| \to \infty \cdot \)

Emmanuele DiBenedetto

XII. Non-negative solutions in ∑ T The case 1

Abstract

We will investigate the structure of non-negative solutions in the stripETof the singular p.d.e

$$ u_t - div|Du|^{p - 2} Du = 0, 1 < p < 2. $$

(1.1)

A striking feature of these singular equations is that, unlike the degenerate casep>2, non-negative solutions of (1.1) are not restricted by any ‘growth condition’ as |x|→∞ .Nevertheless they have initial traces that are Radon measures. Moreover they areuniquewhenever the initial traces are in L¹ _loc (R^N).

Emmanuele DiBenedetto

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