Finite-Trace Strong Solutions of Quadratic-Gradient Parabolic Equations: A Matrix-Valued Decay Estimate and a Clarification of Maz’ya’s Problem 55

1. Introduction

Maz’ya’s Problem 55 concerns the large-time behavior of

$$u_t-\Delta u=\nabla u^{\mathsf{T}}A\left(x\right)\nabla u\mathrm{in}{Q}_{\infty }:=\Omega \times (0,\infty ),$$

(1)

subject to

$$u=0\mathrm{on}\partial \Omega \times (0,\infty ).$$

(2)

The printed problem first assumes that A is positive definite. In the unit-matrix case it invokes the logarithmic substitution and states an alternative between exponential vanishing and behavior linear in time away from the boundary. It then asks for the corresponding result for general A, for the asymptotics of unbounded solutions, and for the non-positive-definite case [12] [Problem 55].

Relation to Prior Scalar Results

The scalar isotropic equation

$$u_t-\Delta u=a{|\nabla u|}^p$$

(3)

with homogeneous Dirichlet data was studied in detail by Benachour, Dăbuleanu-Hapca, and Laurençot [1]. For $p>1$, they proved that global classical solutions converge to zero in $W^{1,\infty }(\Omega )$ at the same exponential rate as the linear Dirichlet heat semigroup. In particular, for $p=2$ they used the Cole–Hopf transform

$$U=e^{au}-1$$

to obtain an exact heat-semigroup representation, and they also identified first-eigenfunction large-time behavior. Consequently, neither scalar quadratic decay nor the Cole–Hopf formula is claimed as new here.

The point specific to the present note is different. We give a direct two-sided comparison argument for the matrix-valued equation (1), allowing $A=A\left(x\right)$ to be merely bounded and measurable. The symmetric part may be anisotropic and indefinite. The estimate is pointwise and weighted by the first Dirichlet eigenfunction, and it can be started from any finite Lipschitz time slice. Since

$${\xi }^{\mathsf{T}}A\xi ={\xi }^{\mathsf{T}}{\displaystyle \frac{A+A^{\mathsf{T}}}{2}}\xi ,$$

the skew-symmetric part is algebraically irrelevant; the substantive extension concerns spatial dependence, anisotropy, and indefiniteness of the symmetric part.

Scope of the Result

There is a basic distinction hidden in the formulation of Problem 55. Under the standard finite Cauchy–Dirichlet interpretation, one prescribes finite data and requires (2) to be attained as an ordinary trace. Under singular or generalized interpretations, a solution may diverge at the boundary, fail to attain the boundary datum in the classical sense, or be governed by a state-constraint or ergodic problem. These classes have different asymptotic mechanisms.

This paper proves a conditional decay theorem for global finite strong solutions in the first class. It does not prove existence, uniqueness, global continuation, regularization from rough data, or a classification of singular solutions. Within the stated class, however, the estimate rules out every nonzero linear-in-time branch while the finite zero trace is maintained.

The distinction between finite and singular boundary behavior is consistent with the wider theory of viscous Hamilton–Jacobi equations. Generalized Dirichlet conditions, state constraints, unbounded stationary solutions, ergodic constants, and boundary blow-up profiles are treated in, among others, [2,3,4,9,10,14]. Those works provide conceptual background rather than a direct theorem for the present quadratic matrix equation.

The results established below are:

(i)

a conditional first-eigenfunction-weighted decay estimate for bounded spatially dependent matrix fields;

(ii)

a proof that uses only the operator-norm bound of the symmetric part and hence does not require positivity or ellipticity;

(iii)

a precise separation between the finite-trace regime and the standard logarithmically singular profile producing linear temporal behavior.

2. Setting and Preliminary Estimates

Throughout, $\Omega \subset {\mathbb{R}}^n$ is bounded, connected, and of class $C^2$. Let ${\lambda }_1>0$ be the first Dirichlet eigenvalue of $-\Delta$, and let ${\varphi }_1$ be the corresponding eigenfunction normalized by

$$-\Delta {\varphi }_1={\lambda }_1{\varphi }_1\mathrm{in}\Omega ,{\varphi }_1=0\mathrm{on}\partial \Omega ,{\varphi }_1>0\mathrm{in}\Omega ,{∥{\varphi }_1∥}_{L^{\infty }(\Omega )}=1.$$

(4)

Denote by $S_D\left(t\right)=e^{t{\Delta }_D}$ the Dirichlet heat semigroup. We use standard parabolic comparison and regularity results as in [11], and standard spectral and positivity properties of Dirichlet heat semigroups as in [6,13].

Definition 2.1

(Finite strong Cauchy–Dirichlet solution). Fix $t_0\ge 0$ and $p>n+2$. A function u is called a finite strong solution of (1)–(2) on $[t_0,\infty )$ if

$$u\in C(\overline{\Omega }\times [t_0,\infty )),u\in W_p^{2,1}(\Omega \times (t_0,T))\mathrm{for}\mathrm{every}T>t_0,$$

the equation holds almost everywhere, and $u=0$ continuously on $\partial \Omega \times [t_0,\infty )$. We write $u_0:=u(·,t_0)$.

For a measurable function f satisfying $\left|f\right|\le C{\varphi }_1$ almost everywhere, define the weighted norm

$${∥f∥}_{{\varphi }_1,\infty }:=\underset{x\in \Omega }{ess\; sup}{\displaystyle \frac{\left|f\right(x\left)\right|}{{\varphi }_1\left(x\right)}}.$$

(5)

The next two elementary facts make the analytic framework of the transformation argument explicit.

Lemma 2.2

(Sobolev chain rule on finite cylinders). Let $T>t_0$, $p>n+2$, and $u\in W_p^{2,1}(\Omega \times (t_0,T))$. If $F\in C^2\left(\mathbb{R}\right)$, then $F\left(u\right)\in W_p^{2,1}(\Omega \times (t_0,T))$ and, almost everywhere,

$$\begin{array}{cc}\hfill {\partial }_{t}F\left(u\right)& =F^{\prime }\left(u\right)u_t,\hfill \\ \hfill \nabla F\left(u\right)& =F^{\prime }\left(u\right)\nabla u,\hfill \\ \hfill \Delta F\left(u\right)& =F^{\prime }\left(u\right)\Delta u+F^{\prime \prime }\left(u\right){|\nabla u|}^2.\hfill \end{array}$$

Proof. 

Parabolic Sobolev embedding for $p>n+2$ gives bounded representatives of u and $\nabla u$ on the finite cylinder. Hence $F^{\prime }$, $F^{\prime \prime }$ are bounded on the range of u. The standard first- and second-order Sobolev chain rules, applied in space and time, give the asserted membership and identities. In particular, the quadratic term $F^{\prime \prime }\left(u\right){|\nabla u|}^2$ belongs to $L^p$. □

Lemma 2.3

(Weak comparison on a finite cylinder). Let $T>t_0$ and let

$$w\in W_p^{2,1}(\Omega \times (t_0,T))\cap C(\overline{\Omega }\times [t_0,T]),p>n+2.$$

If

$$({\partial }_t-\Delta )w\le 0\mathrm{a}.\mathrm{e}.,$$

and $w\le 0$ on the parabolic boundary

$$(\overline{\Omega }\times \left\{t_0\right\})\cup (\partial \Omega \times [t_0,T]),$$

then $w\le 0$ in $\overline{\Omega }\times [t_0,T]$.

Proof. 

The positive part $w_{+}$ has zero lateral trace and vanishes at $t=t_0$. Testing the differential inequality by $w_{+}$, justified for instance by Steklov averaging in time, gives for almost every $\tau \in (t_0,T)$

$${\displaystyle \frac{1}{2}}{∥w_{+}(·,\tau )∥}_{L^2(\Omega )}^2+{\int }_{t_0}^{\tau }{\int }_{\Omega }{|\nabla w_{+}|}^{2}dxdt\le 0.$$

Thus $w_{+}=0$ almost everywhere, and continuity yields the pointwise conclusion. □

The condition $p>n+2$ is therefore used both for the chain rule and for continuous parabolic traces. The argument also applies directly to classical solutions when A is continuous.

We now record the weighted semigroup estimate used below.

Lemma 2.4

(First-eigenfunction domination). If $f\in W^{1,\infty }(\Omega )\cap C\left(\overline{\Omega }\right)$ and $f=0$ on $\partial \Omega$, then

$$|S_D\left(t\right)f\left(x\right)|\le {∥f∥}_{{\varphi }_1,\infty }{e}^{-{\lambda }_{1}t}{\varphi }_1\left(x\right)(x\in \Omega ,t\ge 0).$$

(6)

The quotient is understood in the interior; its essential supremum is finite.

Proof. 

By the Hopf boundary lemma and compactness of $\partial \Omega$, there are $c_1,c_2>0$ such that

$$c_1(x,\partial \Omega )\le {\varphi }_1\left(x\right)\le c_2(x,\partial \Omega )$$

near the boundary. Since f is Lipschitz and has zero trace,

$$\left|f\left(x\right)\right|\le {∥\nabla f∥}_{L^{\infty }}(x,\partial \Omega ).$$

On compact subsets of $\Omega$, ${\varphi }_1$ has a positive minimum. Hence $\left|f\right|\le M_f{\varphi }_1$ in $\Omega$, where $M_f={∥f∥}_{{\varphi }_1,\infty }$. Positivity of $S_D\left(t\right)$ and (4) give

$$|S_D\left(t\right)f|\le S_D\left(t\right)\left|f\right|\le M_f{S}_D\left(t\right){\varphi }_1=M_f{e}^{-{\lambda }_{1}t}{\varphi }_1.$$

□

Remark 2.5.

The Lipschitz assumption on f is a convenient sufficient condition. The proof only requires $\left|f\right|\le C{\varphi }_1$. Thus the main theorem remains valid for any finite time slice for which the two transformed data satisfy this weighted boundary condition.

3. Exponential Decay for Bounded Matrix Fields

For a matrix field A, write

$$A_s\left(x\right):={\displaystyle \frac{A\left(x\right)+A{\left(x\right)}^{\mathsf{T}}}{2}}.$$

The skew-symmetric part does not contribute:

$$\nabla u^{\mathsf{T}}A\left(x\right)\nabla u=\nabla u^{\mathsf{T}}{A}_s\left(x\right)\nabla u.$$

(7)

Set

$$K:=\underset{x\in \Omega }{ess\; sup}{∥A_s\left(x\right)∥}_{\mathrm{op}}.$$

(8)

Theorem 3.1

(Finite-data decay). Let $A\in L^{\infty }(\Omega ;{\mathbb{R}}^{n\times n})$, without any symmetry or definiteness assumption, and let u be a finite strong solution in the sense of Theorem 2.1. Assume

$$u_0\in W^{1,\infty }(\Omega )\cap C\left(\overline{\Omega }\right),u_0=0\mathrm{on}\partial \Omega .$$

Then, for every $x\in \Omega$ and $t\ge t_0$,

$$\left|u(x,t)\right|\le C_0{e}^{-{\lambda }_1(t-t_0)}{\varphi }_1\left(x\right),$$

(9)

where one may take

$$C_0=e^{K{∥u_0∥}_{L^{\infty }(\Omega )}}{∥u_0∥}_{{\varphi }_1,\infty }.$$

(10)

Consequently,

$${∥u(·,t)∥}_{L^{\infty }(\Omega )}\le C_0{e}^{-{\lambda }_1(t-t_0)}.$$

(11)

Proof. 

Fix an arbitrary $T>t_0$ and work first on $\Omega \times (t_0,T)$. If $K=0$, then (7) implies that u solves the linear heat equation, and (9) follows directly from Theorem 2.4. Assume henceforth that $K>0$.

Define

$$E_{+}(x,t):=e^{Ku(x,t)},E_{-}(x,t):=e^{-Ku(x,t)}.$$

By Theorem 2.2, the functions $E_{\pm }$ belong to $W_p^{2,1}$ on the finite cylinder and satisfy, almost everywhere,

$$\begin{array}{cc}\hfill ({\partial }_t-\Delta )E_{+}& =K{e}^{Ku}\left(u_t-\Delta u-K{|\nabla u|}^2\right)\hfill \\ \hfill & =K{e}^{Ku}\left(\nabla u^{\mathsf{T}}{A}_s\nabla u-K{|\nabla u|}^2\right)\le 0,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill ({\partial }_t-\Delta )E_{-}& =-K{e}^{-Ku}\left(u_t-\Delta u+K{|\nabla u|}^2\right)\hfill \\ \hfill & =-K{e}^{-Ku}\left(\nabla u^{\mathsf{T}}{A}_s\nabla u+K{|\nabla u|}^2\right)\le 0.\hfill \end{array}$$

Thus $E_{+}$ and $E_{-}$ are subcaloric. Since $u=0$ on the lateral boundary,

$$E_{+}=E_{-}=1\mathrm{on}\partial \Omega \times [t_0,\infty ).$$

For $t\ge t_0$, let

$$H_{\pm }(x,t):=1+S_D(t-t_0)\left(e^{\pm K{u}_0}-1\right)\left(x\right).$$

(12)

Each $H_{\pm }$ is caloric for $t>t_0$, extends continuously to $t=t_0$, has lateral boundary value one, and agrees there with $E_{\pm }$. To avoid imposing a maximal-regularity compatibility condition on the merely Lipschitz initial data, fix $\epsilon >0$. On $[t_0+\epsilon ,T]$, the differences $E_{\pm }-H_{\pm }$ belong to $W_p^{2,1}$ and vanish on the lateral boundary. Put

$${\delta }_{\pm }\left(\epsilon \right):=max\left\{0,\underset{x\in \overline{\Omega }}{sup}(E_{\pm }-H_{\pm })(x,t_0+\epsilon )\right\}.$$

Continuity at $t=t_0$ gives ${\delta }_{\pm }\left(\epsilon \right)\to 0$ as $\epsilon \downarrow 0$. Applying Theorem 2.3 to $E_{\pm }-H_{\pm }-{\delta }_{\pm }\left(\epsilon \right)$ on the shortened cylinder and then letting $\epsilon \downarrow 0$ yields

$$E_{+}(x,t)\le H_{+}(x,t),E_{-}(x,t)\le H_{-}(x,t).$$

(13)

Set $f_{\pm }=e^{\pm K{u}_0}-1$. These functions are Lipschitz and vanish on $\partial \Omega$. By Theorem 2.4,

$$|H_{\pm }(x,t)-1|\le M_{\pm }{e}^{-{\lambda }_1(t-t_0)}{\varphi }_1\left(x\right),M_{\pm }:={∥e^{\pm K{u}_0}-1∥}_{{\varphi }_1,\infty }.$$

(14)

At a point where $u(x,t)\ge 0$, (13) gives

$$1\le e^{Ku(x,t)}\le H_{+}(x,t).$$

Hence $H_{+}(x,t)\ge 1$, and $logy\le y-1$ for $y\ge 1$ gives

$$0\le u(x,t)\le {\displaystyle \frac{H_{+}(x,t)-1}{K}}\le {\displaystyle \frac{M_{+}}{K}}{e}^{-{\lambda }_1(t-t_0)}{\varphi }_1\left(x\right).$$

At a point where $u(x,t)\le 0$, the same argument applied to $E_{-}$ gives

$$0\le -u(x,t)\le {\displaystyle \frac{M_{-}}{K}}{e}^{-{\lambda }_1(t-t_0)}{\varphi }_1\left(x\right).$$

Finally, the mean-value theorem implies

$$|e^{\pm Ks}-1|\le K{e}^{K\left|s\right|}\left|s\right|,$$

so that

$${\displaystyle \frac{1}{K}}max\{M_{+},M_{-}\}\le e^{K{∥u_0∥}_{L^{\infty }}}{∥u_0∥}_{{\varphi }_1,\infty }=C_0.$$

This proves (9) on $[t_0,T]$. Since $T>t_0$ was arbitrary, the estimate holds for all $t\ge t_0$; (11) follows from ${∥{\varphi }_1∥}_{\infty }=1$. □

Remark 3.2 (One regular time slice is enough)

No regularity assumption at the original initial time is needed. If a global solution has a single time $t_0>0$ at which $u(·,t_0)$ is finite, Lipschitz, and has zero trace, then Theorem 3.1 applies from that time onward.

Corollary 3.3

(No linear branch in the finite class). Under the hypotheses of Theorem 3.1, $u(·,t)\to 0$ uniformly and exponentially. In particular, no solution can satisfy

$$u(x,t)=ct+O\left(1\right)$$

with $c\ne 0$ at any fixed interior point, or locally uniformly on any nonempty open subset of Ω.

Corollary 3.4

(Rigidity of finite stationary states). Let $A\in L^{\infty }(\Omega ;{\mathbb{R}}^{n\times n})$. If

$$U\in W^{2,p}(\Omega )\cap W^{1,\infty }(\Omega )\cap C\left(\overline{\Omega }\right),p>n,$$

satisfies

$$-\Delta U=\nabla U^{\mathsf{T}}A\left(x\right)\nabla U\mathrm{a}.\mathrm{e}.\mathrm{in}\Omega ,U=0\mathrm{on}\partial \Omega ,$$

then $U\equiv 0$.

Proof. 

Since $p>n$, Sobolev embedding gives $\nabla U\in L^{\infty }(\Omega )$. Hence

$$F:=\nabla U^{\mathsf{T}}A\nabla U\in L^{\infty }(\Omega ).$$

The global $W^{2,q}$ regularity theorem for the homogeneous Dirichlet problem on a $C^2$ domain [7] applied to $-\Delta U=F$ gives

$$U\in W^{2,q}(\Omega )\mathrm{for}\mathrm{every}\mathrm{finite}q.$$

Choose $q>n+2$. The time-independent function $u(x,t):=U\left(x\right)$ then belongs to $W_q^{2,1}(\Omega \times (t_0,T))$ for every finite cylinder, has the required continuous zero trace, and is a finite strong solution in the sense of Theorem 2.1. Applying Theorem 3.1 and letting $t\to \infty$ gives $\left|U\right(x\left)\right|\le 0$, hence $U\equiv 0$. □

Remark 3.5 (Optimality of the uniform exponent)

The exponent ${\lambda }_1$ is optimal as a rate uniform over the stated class. For $A=0$,

$$u(x,t)=e^{-{\lambda }_1(t-t_0)}{\varphi }_1\left(x\right)$$

is an exact solution. This does not assert that every individual solution has a nonzero first-mode coefficient; particular data may decay faster.

4. The Scalar Quadratic Case: A Known Formula and a Spectral Consequence

For comparison with the matrix-valued estimate, we recall the scalar quadratic transform. This representation is classical and was used explicitly for the homogeneous Dirichlet problem by Benachour, Dăbuleanu-Hapca, and Laurençot [1] [Remark 1.2 and formula (4.11)]; it is not claimed as a new contribution here. When $A=I$, put

$$v=e^u.$$

Then

$$v_t-\Delta v=e^u\left(u_t-\Delta u-{|\nabla u|}^2\right)=0,$$

and the finite boundary condition $u=0$ becomes $v=1$. Therefore, for $u_0=u(·,t_0)$,

$$v(x,t)=1+S_D(t-t_0)(e^{u_0}-1)\left(x\right),u(x,t)=log\left(1+S_D(t-t_0)(e^{u_0}-1)\left(x\right)\right).$$

(15)

Positivity of v follows either from $v=e^u$ or directly from the heat-equation maximum principle.

For completeness, the known exact formula also gives an explicit leading coefficient. Let ${\lambda }_2>{\lambda }_1$ denote the second distinct Dirichlet eigenvalue, and set

$$f:=e^{u_0}-1,a_1:={\displaystyle \frac{{\int }_{\Omega }f{\varphi }_{1}dx}{{\int }_{\Omega }{\varphi }_1^{2}dx}}.$$

(16)

Proposition 4.1

(Explicit spectral consequence for $A=I$). Assume $u_0\in W^{1,\infty }(\Omega )\cap C\left(\overline{\Omega }\right)$ with zero boundary trace. Then, as $t\to \infty$,

$$u(·,t)=a_1{e}^{-{\lambda }_1(t-t_0)}{\varphi }_1+O\left(e^{-\mu (t-t_0)}\right)\mathrm{in}{L}^{\infty }(\Omega ),$$

(17)

where

$$\mu :=min\{{\lambda }_2,2{\lambda }_1\}>{\lambda }_1.$$

In particular, if $a_1\ne 0$, the first Dirichlet mode gives the exact leading order. If $a_1=0$, the solution decays strictly faster than $e^{-{\lambda }_{1}t}$.

Proof. 

The Dirichlet Laplacian is self-adjoint with compact resolvent. Spectral decomposition gives

$$S_D\left(s\right)f=a_1{e}^{-{\lambda }_{1}s}{\varphi }_1+R\left(s\right),s=t-t_0,$$

where ${∥R\left(s\right)∥}_{L^2}=O\left(e^{-{\lambda }_{2}s}\right)$. For $s\ge 1$, write $R\left(s\right)=S_D\left(1\right)R(s-1)$. The standard ultracontractive smoothing bound $S_D\left(1\right):L^2(\Omega )\to L^{\infty }(\Omega )$ [6,13] yields

$${∥R\left(s\right)∥}_{L^{\infty }}=O\left(e^{-{\lambda }_{2}s}\right).$$

Moreover, Theorem 2.4 gives ${∥S_D\left(s\right)f∥}_{\infty }=O\left(e^{-{\lambda }_{1}s}\right)$. For all sufficiently large s, ${∥S_D\left(s\right)f∥}_{\infty }\le 1/2$. Hence $log(1+z)=z+O\left(z^2\right)$ uniformly for $\left|z\right|\le 1/2$, and formula (15) implies

$$u=S_D\left(s\right)f+O\left({∥S_D\left(s\right)f∥}_{\infty }^2\right),$$

which is (17). □

5. Where the Linear Solution Belongs

The unit-matrix case also displays the boundary mismatch directly. Let

$$v(x,t)=e^{-{\lambda }_{1}t}{\varphi }_1\left(x\right),u(x,t)=logv(x,t)=-{\lambda }_{1}t+log{\varphi }_1\left(x\right).$$

(18)

Then $v_t-\Delta v=0$, so u solves

$$u_t-\Delta u={|\nabla u|}^2\mathrm{in}\Omega \times (0,\infty ).$$

For every compact $K⋐\Omega$,

$$u(x,t)=-{\lambda }_{1}t+O_K\left(1\right),$$

which is a genuine linear-in-time regime. However,

$$u(x,t)⟶-\infty \mathrm{as}x\to \partial \Omega ,$$

(19)

because ${\varphi }_1=0$ on $\partial \Omega$. Thus (18) does not satisfy (2) as a finite classical trace.

More generally, for $A=aI$ with $a\ne 0$, the transform $v=e^{au}$ is caloric and

$$u(x,t)={\displaystyle \frac{1}{a}}\left(-{\lambda }_{1}t+log{\varphi }_1\left(x\right)\right)$$

solves the equation in the interior. Its boundary trace is infinite, with sign determined by a.

This is not a minor technical defect. It identifies two different boundary-value problems:

(a)

finite Cauchy–Dirichlet data, for which Theorem 3.1 forces exponential convergence to zero;

(b)

singular, generalized, or state-constraint data, where boundary blow-up and ergodic behavior may occur.

The second class is related to the viscosity and large-solution theories developed in [2,3,4,9,10,14]. In particular, the literature on viscous Hamilton–Jacobi equations distinguishes convergence to stationary Dirichlet solutions from asymptotics involving an ergodic constant and a singular stationary profile. Those results illustrate why the singular branch must be formulated as a separate boundary-value problem; they do not by themselves establish the asymptotics of the present matrix equation.

6. A Precise Interpretation of Problem 55

The preceding analysis leads to the following statement.

Proposition 6.1

(Consequence within the finite-trace class). Under finite strong Cauchy–Dirichlet data, the exponential-decay branch is the only possible branch. This remains true for every bounded matrix field A, whether or not A is symmetric or positive definite. Consequently, any solution with nonzero linear behavior in time must leave the class of Theorem 2.1, for example through an infinite boundary trace, failure to attain the prescribed boundary data, or a non-finite time slice.

The proposition gives a conditional answer within the finite-trace strong-solution class, including indefinite bounded matrix fields, but it does not answer the request to “describe an asymptotic behaviour of unbounded solutions.” The term “unbounded” may refer to growth in time, blow-up toward the boundary, or a generalized solution that does not attain its boundary datum. These possibilities require a specified solution concept and boundary interpretation.

A natural formal ansatz is

$$u(x,t)=-ct+\psi \left(x\right)+o\left(1\right).$$

(20)

Substitution into (1) gives the nonlinear additive-eigenvalue problem

$$-\Delta \psi -\nabla {\psi }^{\mathsf{T}}A\left(x\right)\nabla \psi =c\mathrm{in}\Omega .$$

(21)

For $A=I$,

$$\psi =log{\varphi }_1,c={\lambda }_1,$$

solves (21) and satisfies $\psi \to -\infty$ at the boundary. This is the exact stationary profile behind (18).

For general matrix fields, the existence, uniqueness, sign, and boundary asymptotics of singular pairs $(c,\psi )$ for (21) form a separate nonlinear eigenvalue problem. Positive definiteness, anisotropy, nonsymmetry (through the irrelevant skew part), and indefiniteness may have genuinely different effects there. A complete treatment would need to specify, at minimum:

(i)

the admissible singular boundary behavior;

(ii)

the solution concept for (21);

(iii)

the normalization and uniqueness criterion for $\psi$;

(iv)

the class of parabolic solutions for which (20) is to be proved.

These questions are outside the scope of the present note.

7. Conclusion

Assuming the existence of a global finite strong solution with one Lipschitz time slice, the two exponential transforms $e^{\pm Ku}$ yield the pointwise estimate

$$\left|u(x,t)\right|\le C_0{e}^{-{\lambda }_1(t-t_0)}{\varphi }_1\left(x\right).$$

The proof uses only a bound on the symmetric part of A and therefore applies to bounded spatially dependent, anisotropic, and indefinite matrix fields. No existence, uniqueness, or continuation theorem is contained in this argument.

For the scalar equation $u_t-\Delta u=a{|\nabla u|}^2$, decay at the heat-semigroup rate and the Cole–Hopf representation were already established by Benachour, Dăbuleanu-Hapca, and Laurençot [1]. The present matrix comparison estimate should be read as an extension of that finite-data mechanism, not as a new proof of the scalar theory.

The profile $-{\lambda }_{1}t+log{\varphi }_1$ demonstrates why the linear temporal branch requires a different boundary interpretation: its trace is $-\infty$, not the finite value zero. Accordingly, the paper resolves only a finite-trace strong-solution subcase of Maz’ya’s Problem 55. The existence and asymptotic classification of unbounded or boundary-singular solutions remain separate open questions requiring a specified generalized solution concept.