Existence of Non-Negative Solutions for Parabolic Problem on Riemannian Manifold

Abstract

In this paper, we investigate a perturbed parabolic problem involving the Laplace–Beltrami operator on a smooth compact Riemannian manifold M. For a strongly local Dirichlet form in $L^2\left(M\right)$. More precisely, we begin by proving that, in the case of the existence of a non-negative solution, the potential can be written as a derivative of some functions which are locally integrable on M; after that, we prove the existence of a non-negative solution for such problems.

1. Introduction

The study of parabolic problems on Riemannian manifolds lies at the intersection of differential equations, geometry, and analysis. Parabolic problems typically arise from heat equations, which describe diffusion processes. On Riemannian manifolds, these problems are naturally formulated in the context of geometric analysis, where the manifold is endowed with a Riemannian metric that influences the behavior of solutions.

A parabolic problem typically refers to a partial differential equation with a time evolution that is governed by a second-order differential operator, such as the heat operator. In Euclidean space, the simplest form is the heat equation:

$${\displaystyle \frac{\partial u}{\partial t}}=\Delta u,$$

where $u(x,t)$ is the unknown function, and $\Delta$ is the Laplacian. These problems have applications in various areas of mathematics and physics, especially in the study of geometric evolution, diffusion processes, and analysis on curved spaces.

On a Riemannian manifold $(M,g)$, where g is the Riemannian metric, the heat equation is generalized to the following:

$${\displaystyle \frac{\partial u}{\partial t}}={\Delta }_{g}u,$$

where ${\Delta }_g$ is the Laplace–Beltrami operator, a generalization of the Laplace operator to curved spaces. The Laplace–Beltrami operator is defined in terms of the metric g and involves the covariant derivatives and the curvature of the manifold.

In this paper, we consider a smooth compact manifold $(M^n,g)$, $n\ge 3$, and we discuss the existing results of non-negative solutions for the following perturbed heat equation:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{\partial u}{\partial t}}=({\Delta }_{\mu }+V)u,\hfill & \mathrm{in}M\times (\delta ,T),0<\delta <T\le +\infty ,\hfill \\ u(x,\delta )=u_{\delta }\left(x\right),\hfill & x\in M,\hfill \end{array}\right.\end{array}$$

(1)

where $\mu$ is a Radon measure on M with full support, and ${\Delta }_{\mu }$ is the Laplace–Beltrami operator on M, $u_{\delta }\in L^2(M,\mu )$, $V\in L^2(M\times [\delta ,T],d\mu dt)$.

Our motivation for looking at this problem comes from the work of [1], in which the authors considered the following parabolic problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\Delta u+Vu-{\displaystyle \frac{\partial u}{\partial t}}=0,\hfill & \mathrm{in}{\mathbb{R}}^n\times (0,\infty ),\hfill \\ u(x,0)=u_0\left(x\right),on{\mathbb{R}}^n,\hfill \end{array}\right.\end{array}$$

(2)

where $u_0\in L^2\left({\mathbb{R}}^n\right)$ and $V\in L^2({\mathbb{R}}^n\times (0,\infty )),n\ge 3$. Since ${\mathbb{R}}^n$ is a particular case of the Riemann manifold, our paper generalizes the work of Zhang [1]. Ghanmi and Kenzizi [2] studied the following parabolic problem:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\Delta u+Vu-{\displaystyle \frac{\nabla \phi }{\phi }}\nabla u-{\displaystyle \frac{\partial u}{\partial t}}=0,\hfill & \mathrm{in}D\times (0,\infty ),\hfill \\ \phi \left(x\right)u(x,t)=0,\hfill & \mathrm{on}\partial D\times (0,\infty ),\hfill \\ u(x,0)=u_0\left(x\right),\hfill & \mathrm{in}D,\hfill \end{array}\right.\end{array}$$

(3)

where $\phi$ is the normalized positive eigenfunction corresponding to the first eigenvalue of $-\Delta$, and V belongs to a class of time-dependent potentials. The novelty of our paper, compared with the last work, is that we consider more classes of functions; moreover, these functions are assumed to be locally integrable. For interested readers about a class of potentials that can be written as the spatial derivative of some functions, we refer them to [3,4,5]).

We note that the parabolic equation is often used to model heat diffusion, and this can be extended to curved spaces. Understanding how heat spreads over a Riemannian manifold is crucial for various problems in thermodynamics, especially in the context of complex geometries. One of the most famous applications of parabolic equations on Riemannian manifolds is in the Ricci flow. Also, parabolic equations on manifolds are used in fields like image processing, machine learning, and models of biological diffusion processes; for interested readers, we refer them to [6,7,8,9].

In this paper, we propose generalizing the results in [1]. To this aim, let us consider the case of the Laplace–Beltrami operator on M associated with Dirichlet form $\mathcal{E}$ via Kato’s representation theorem [10], that is,

$$(-{\Delta }_{\mu }u,v)=\mathcal{E}(u,v).$$

(4)

From the physical perspective, such quadratic forms may intuitively be understood as a model of functional energy. The measured perturbation of Dirichlet forms have been studied in generally increasing increments and with different aims. The purpose of this paper is to adapt the strategy from [1,2,11,12,13,14] to a local Dirichlet form given by (4).

To outline our main results, it is convenient to introduce the following notations.

A Riemannian metric (or a Riemannian metric tensor) on a smooth n-$dimensional$ manifold M is a family $g={\left\{g\left(x\right)\right\}}_{x\in M}$ such that $T_{x}M$, smoothly depending on $x\in M$. Using the metric tensor, one defines an inner product ${\langle .,.\rangle }_g$ in any tangent space $T_{x}M$ by

$${\langle \xi ,\eta \rangle }_g\equiv g\left(x\right)(\xi ,\eta ),$$

for all tangent vectors $\xi ,\eta \in T_{x}M$. Hence, $T_{x}M$ becomes an Euclidean space.

Assume in the sequel that $(M,g,\mu )$ is a weighted manifold. The couple $(M,\mu )$ can also be considered as a measure space. Hence, the notions of measurable and integrable functions are defined as well as the Lebesgue function spaces $L^p\left(M\right)=L^P(M,\mu ),1\le p\le +\infty$.

The space of all measurable functions f on M is denoted as $L_{loc}^p\left(M\right)$ such that $f\in L^p\left(\mathrm{\Omega }\right)$ for any relatively compact open set $\mathrm{\Omega }\subset M$. We now define the following Sobolev space

$$W^1\left(M\right)=W^1(M,g,\mu ):=\{u\in L^2\left(M\right):\nabla u\in {\overrightarrow{L}}^2\left(M\right)\},$$

where the space ${\overrightarrow{L}}^2\left(M\right)$ consists of measurable vector fields v such that $\left|v\right|\in L^2\left(M\right)$ (where $\left|v\right|={\langle v,v\rangle }_g^{{\displaystyle \frac{1}{2}}}$ is the length of v).

In addition, we define the space $W_{loc}^1\left(M\right)$ by

$$W_{loc}^1\left(M\right)=\left\{u\in L_{loc}^2\left(M\right),\nabla u\in {\overrightarrow{L}}_{loc}^2\left(M\right)\right\}.$$

Let us recall two definitions we need in the rest of this paper.

Definition 1.

Let $\alpha >0$, $T>\delta$, and U be a chart in M. Given two compactly supported functions, $V\in L^2(U\times [\delta ,T],d\mu dt)$ and $f\in L_{loc}^1(U\times [\delta ,T],d\mu dt)$, we formulate the following:

$$V={\Delta }_{\mu }f-\alpha {\left|{\nabla }_{g}f\right|}_g^2-{\displaystyle \frac{\partial f}{\partial t}}.$$

If there exist sequences of functions $\left\{V_k\right\}$ and $\left\{f_k\right\}$ such that, for all $k=1,2,\cdots$, the following conditions hold:

1. 

$V_k\in L^2\left(U\times [\delta ,T],d\mu dt\right),f_k\in L^2\left(U\times [\delta ,T],d\mu dt\right),{\Delta }_{\mu }{f}_k\in L^2\left(U\times [\delta ,T],d\mu dt\right),$${\partial }_t{f}_k\in L^2\left(U\times [\delta ,T],d\mu dt\right).$

2. 

$V_k\to Vin{L}^2(U\times [\delta ,T],d\mu dt),|V_i|\le |V_{i+1}|,f_k\to f\mu -a.e.and{f}_k(x,\delta )=f_{k+1}(x,\delta ).$

3. 

$V_k={\Delta }_{\mu }{f}_k-\alpha {\left|{\nabla }_g{f}_k\right|}_g^2-{\displaystyle \frac{\partial f_k}{\partial t}}.$

Definition 2.

Let $V\in L^2(M\times [\delta ,T])$, $u_{\delta }\in L^2\left(M\right)$. We say that a Borel measurable function $u:M\times [\delta ,T]\to \mathbb{R}$ is a solution of the heat equation as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{\partial u}{\partial t}}=({\Delta }_{\mu }+V)u,\hfill & inM\times (\delta ,T),\hfill \\ u(x,\delta )=u_{\delta }\left(x\right),\hfill & fora.e.x\in M,\hfill \end{array}\right.\end{array}$$

(5)

if

1. 

$u\in L_{loc}^2([\delta ,T],W^1\left(M\right))$.

2. 

$uV\in L_{loc}^1(M\times (\delta ,T),d\mu dt)$.

3. 

For every $0<\delta \le t<T$ and every Borel function $\phi :M\times [\delta ,T]\to \mathbb{R}$ such that

$${\displaystyle \phi ,\frac{\partial \phi }{\partial t}\in L^2(M\times (\delta ,T)),\phi (.,t)\in \mathcal{D}om\left({\Delta }_{\mu }\right),and{\int }_{\delta }^t{\int }_{M}u(s,x){\Delta }_{\mu }\phi d\mu ds<\infty }$$

and the following identity holds:

$$\begin{array}{cc}\hfill {\int }_{\delta }^t{\int }_{M}Vu(s,x)\phi (s,x)d\mu ds& {\displaystyle ={\int }_{\delta }^t{\int }_M(\left(u\phi \right)(t,,x)-u_{\delta }\left(x\right)\phi (0,x))d\mu ds}\hfill \\ & {\displaystyle -{\int }_{\delta }^t{\int }_{M}u(s,x)({\Delta }_{\mu }\phi (s,x)+\frac{\partial \phi }{\partial s}(s,x))d\mu ds.}\hfill \end{array}$$

2. Main Results and Proofs

In this section, we outline our main results and proofs. Our first result is the following:

Theorem 1.

Assume that $u_{\delta }\ge 0$ and (1) has a non-negative solution. Then, there exists f with $e^{-f}\in L^2(M\times [\delta ,T],d\mu dt)$ and $e^{-f(.,t)}\in L^2(M,d\mu )$ such that

$$V={\Delta }_{\mu }f-{\left|{\nabla }_{g}f\right|}_g^2-{\displaystyle \frac{\partial }{\partial t}}f.$$

(6)

Moreover, if $ln\left(u_{\delta }\right)\in L_{loc}^1(M,d\mu )$, then $f\in L_{loc}^1(M\times [\delta ,T],d\mu dt)$.

Proof. 

Assume that $u_{\delta }\ge 0$ and u is a non-negative solution to the problem (1). For $k=1,2,\cdots$, we input $V_k=min(V,k)=V\wedge k$ and denote the heat equation corresponding to the self-adjoint non-negative semidefinite operator associated with the initial Dirichlet form $\mathcal{E}$, perturbed by $(-V_k)$ instead of $(-V)$, as $\left(P_k\right)$. The standard theory of quadratic form [15,16] implies the existence of a unique non-negative energy solution $u_k$ for $\left(P_k\right)$, given by

$$u_k\left(t\right)=e^{-t{L}_k}{u}_{\delta },t>\delta >0,$$

where $L_k$ is the self-adjoint operator associated with the quadratic form $\mathcal{E}-V_k$ defined by

$$D(\mathcal{E}-V_k)=W_0^1\left(M\right)\mathrm{and}(\mathcal{E}-V_k)\left(u\right)=\mathcal{E}(u,u)-{\int }_M{V}_k{u}^{2}d\mu ,$$

(7)

on $L^2\left(M\right)$. The space $W_0^1\left(M\right)$ has the same inner product as $W^1\left(M\right)$ and is a Hilbert space as a closed subspace of $W^1\left(M\right)$. That is, $\mathcal{E}-V_k$ is a closed quadratic form. Furthermore, the solution $u_k$ lies in the space $\mathcal{F}\left(M\right)\cap L^{\infty }(M,\mu )$ with respect to the space variable and is continuous in its time variable. Similar to the arguments used in [17,18], the function $u_k$ satisfies the integral formulation

$$\begin{array}{cc}\hfill u_k(t,x):& =e^{-t{L}_k}{u}_{\delta }\left(x\right)\hfill \\ \hfill & =e^{t{\Delta }_{\mu }}{u}_{\delta }\left(x\right)+{\int }_{\delta }^t{\int }_M{p}_{t-s}(x,y)u_k(y,s)V_k(y,s)d\mu ds,\forall t>0,\hfill \end{array}$$

(8)

where $p_t$, $t>\delta >0$, is the heat kernel of the semigroup $P_t:=e^{t{\Delta }_{\mu }}$. □

We claim that the sequence $\left(u_k\right)$ is increasing. Indeed, using the same idea as in the proof of ([14], Lemma 2.7), we obtained the following:

$$\begin{array}{cc}\hfill {\displaystyle u_{k+1}\left(t\right)-u_k\left(t\right)}& =e^{-t{L}_{k+1}}{u}_{\delta }-e^{-t{L}_k}{u}_{\delta }\hfill \\ \hfill & ={\int }_M[P_{V_{k+1}}(t-\delta ,x-y,0)-P_{V_k}(t-\delta ,x-y,0)]u_{\delta }\left(y\right)d\mu \hfill \\ \hfill & \ge 0,\hfill \end{array}$$

(9)

where $P_{V_k}$ is often referred to as the heat kernel of the corresponding semi-group.

From the definition of the solution, we infer

$$\begin{array}{cc}\hfill {\displaystyle {\int }_{\delta }^t{\int }_M(u_k\left(s\right)-u\left(s\right))(-{\varphi }_s\left(s\right)}& -{\Delta }_{\mu }\varphi \left(s\right)-V_k\varphi \left(s\right))d\mu ds\hfill \\ \hfill & ={\int }_{\delta }^t{\int }_{M}u\varphi (V_k-V)d\mu ds\hfill \\ \hfill & \le 0.\hfill \end{array}$$

(10)

Let $\psi \in C_c^{\infty }(M\times (\delta ,t))$ be non-negative functions such that the following parabolic problem has a positive solution (see [10]):

$$\left\{\begin{array}{cc}-{\displaystyle \frac{\partial \varphi }{\partial s}}=-{\Delta }_{\mu }\varphi +V_k\varphi +\psi ,\hfill & \mathrm{in}M\times (\delta ,t),\hfill \\ \varphi (t,x)=0,\hfill & \forall x\in M,\hfill \end{array}\right.$$

(11)

For fixed t, we have

$$\varphi \left(s\right)={\int }_{\delta }^{t-s}{e}^{-(t-s-\xi )(L-V_k)}\psi (t-\xi )d\xi ,0<\delta \le s\le t,\varphi \left(s\right)=0,\forall s>t.$$

(12)

Combining (10) with (12), we obtain

$${\int }_{\delta }^t{\int }_M(u_k-u)\psi d\mu ds\le 0,\forall 0\le \psi \in C_c^{\infty }((\delta ,t)\times M).$$

(13)

As t is arbitrary, we obtain $u_k\le u$, and consequently,

$$0\le u_{k-1}\le u_k\le u\in L^2([\delta ,T]\times M,d\mu dt)\mathrm{and}u(.,t)\in W^1\left(M\right).$$

It follows from the Dominated Convergence Theorem that there exists h such that

$$h\in L^2(M\times [\delta ,T],d\mu dt)\mathrm{and}{u}_k\to h\mathrm{in}{L}^2([\delta ,T]\times M,d\mu dt).$$

Using again Dominated Convergence Theorem, we obtain

$$V_k{u}_k\to Vhin{L}_{loc}^1(M\times [\delta ,T],d\mu dt)\mathrm{and}Vh\in L_{loc}^1(M\times [\delta ,T],d\mu dt).$$

Hence, h is a non-negative solution of (1) with initial condition $u_0$. Now fixing j and setting $V_{j,k}=V_j\vee (-k)$, $\left(P_{j,k}\right)$ denotes the heat equation corresponding to the self-adjoint semidefinite operator with the initial Dirichlet form $\mathcal{E}$, perturbed by $-V_{j,k}$ instead of $-V$. Using the same argument as in the first part of the proof, we can deduce that $\left(P_{j,k}\right)$ has a unique $L^2$ non-negative solution satisfying $u_{j,k}(.,t)\in W^1\left(M\right)$. Furthermore, we see that ${\left(u_{jk}\right)}_k$ is a non-increasing sequence and $V_{j,k}{u}_{jk}\in L^2([\delta ,T]\times M,d\mu dt),T>0$.

${\mathcal{L}}_{\mu }^{V_{jk}}$ denotes the self-adjoint operator associated with the closed quadratic form ${\mathcal{E}}^{V_{jk}}:=\mathcal{E}-V_{jk}$ defined by

$${\mathcal{E}}^{V_{jk}}:D\left({\mathcal{E}}^{V_{jk}}\right)=dom\left({\mathcal{L}}_{\mu }^{V_{jk}}\right),{\mathcal{E}}^{V_{jk}}\left[f\right]=({\mathcal{L}}_{\mu ,jk}f,f)=\mathcal{E}(f,f)-{\int }_M{f}^2{V}_{jk}d\mu ,$$

and $u_{jk}\left(t\right):=e^{-t{\mathcal{L}}_{\mu }^{V_{jk}}}{u}_{\delta },t\ge \delta$ is the solution of the problem $\left(P_{jk}\right)$.

Now, by using the same argument as in [18] based on the context of the semi-group theory, let $E_{{\lambda }_{jk}}\ge 0$ be the spectral resolution in $L^2\left(M\right)$ of the operator ${\mathcal{L}}_{\mu }^{V_{jk}}$. Then, we have

$$u_{jk}\left(t\right)=e^{-t{\mathcal{L}}_{\mu }^{V_{jk}}}{u}_{\delta }={\int }_0^{+\infty }{e}^{-t{\lambda }_{jk}}d{E}_{{\lambda }_{jk}}{u}_{\delta }$$

and

$${\displaystyle \frac{\partial u_{jk}}{\partial t}}\left(t\right)=-{\mathcal{L}}_{\mu }^{V_{jk}}{e}^{-t{\mathcal{L}}_{\mu }^{V_{jk}}}{u}_{\delta }={\int }_0^{+\infty }(-{\lambda }_{jk})e^{-t{\lambda }_{jk}}d{E}_{{\lambda }_{jk}}{u}_{\delta }.$$

where

$$\begin{array}{cc}\hfill \parallel {\displaystyle \frac{\partial u_{jk}}{\partial t}}{\parallel }_{L^2(M\times (\delta ,T))}^2:& ={\int }_{\delta }^T{\int }_0^{+\infty }{\lambda }_{jk}^2{e}^{-2t{\lambda }_{jk}}d{\parallel E_{{\lambda }_{jk}}{u}_{\delta }\parallel }^{2}dt\hfill \\ \hfill & \le {\displaystyle \frac{T-\delta }{\delta T}}{\parallel u_{\delta }\parallel }_{L^2\left(M\right)}^2.\hfill \end{array}$$

(14)

On the other hand, we have

$$0\le u_{jk}-u_j\le u_{j1}-u_j,k=1,2,3,\cdots .$$

Since, $u_j$ is nondecreasing, we have

$$0\le h-u_j\le h-u_1.$$

Finally, we obtain that $u_{jk}-u_j⟶0,a.e.,ask\to \infty ,$ and $0\le u_{jk}-u_j\le u_{j1}-u_j\in L^2([\delta ,T]\times M,d\mu dt).$ Using the Dominated Convergence Theorem, we have

$${\displaystyle \underset{k\to +\infty }{lim}{\int }_{\delta }^T\parallel (u_{jk}-u_j)(.,t){\parallel }_{W_{loc}^1}^{2}dt=0\mathrm{and}\underset{j\to +\infty }{lim}{\int }_{\delta }^T{\parallel (h-u_j)(.,t)\parallel }_{W_{loc}^1}^{2}dt=0.}$$

Note that $\left(u_j\right)$ is bounded, so we can extract a subsequence also denoted by $\left(u_j\right)$ such that

$${\displaystyle \underset{j\to +\infty }{lim}{\int }_{\delta }^T{\parallel (h-u_j)(.,t)\parallel }_{W_{loc}^1}^{2}dt=0.}$$

Hence, ${\displaystyle \underset{j\to +\infty }{lim}{\parallel h-u_j\parallel }_{L^2}=0,}$ which yields that $\parallel h-u_j{\parallel }_{L^2}\le M.$ Then, there exists a subsequence, still denoted by $\left(u_j\right)$, such that $u_j\to ha.e.$ Recall that $u_{\delta }\ge 0$, $u_{\delta }\ne 0$ and $V_j$ is bounded. Then, $u_j>0ast>0.$ Now, we define the functions $f_k$ and f by

$$f_k(x,t)=-ln{u}_k(x,t)\mathrm{for}\mathrm{all}(x,t)\in M\times (\delta ,T),$$

(15)

and

$$f(x,t)=-lnu(x,t)\mathrm{for}\mathrm{all}(x,t)\in M\times (\delta ,T).$$

(16)

Set $F\left(f_k\right)=u_k=e^{-f_k}$, then ${\displaystyle \frac{\partial }{\partial t}}\left[F\left(f_k\right)\right]=-f_k^{\prime }{e}^{-f_k}$. Let us prove that $V_k={\Delta }_{\mu }{f}_k-{\left|{\nabla }_g{f}_k\right|}_g^2-{\displaystyle \frac{\partial f_k}{\partial t}},a.e$, i.e.,

$${\int }_{\delta }^T\int \left[-{\Delta }_{\mu }{f}_k+{\displaystyle \frac{\partial }{\partial t}}F\left(f_k\right)\right]\phi ={\int }_{\delta }^T\int \left[-{\Delta }_{\mu }F\left(f_k\right)+{\left|{\nabla }_g{f}_k\right|}_g^2+{\displaystyle \frac{\partial }{\partial t}}{f}_k\right]\phi F\left(f_k\right),\forall \phi \in W^1\left(M\right).$$

Note that

$$\begin{array}{cc}\hfill {\displaystyle {\int }_{\delta }^T\int [-{\Delta }_{\mu }F\left(f_k\right)}& +{\displaystyle \frac{\partial }{\partial t}}F\left(f_k\right)]\phi \hfill \\ & ={\int }_{\delta }^T\int -{\Delta }_{\mu }F\left(f_k\right)\phi +{\int }_{\delta }^T\int {\displaystyle \frac{\partial }{\partial t}}\left(f_k\right)F^{\prime }\left(f_k\right)\phi \hfill \\ & ={\int }_{\delta }^T\int {\langle \nabla F\left(f_k\right),\nabla \phi \rangle }_g+{\int }_{\delta }^T\int {\displaystyle \frac{\partial }{\partial t}}\left(f_k\right)F^{\prime }\left(f_k\right)\phi \hfill \\ & ={\int }_{\delta }^T\int F^{\prime }\left(f_k\right){\langle \nabla f_k,\nabla \phi \rangle }_g+{\int }_{\delta }^T\int {\displaystyle \frac{\partial }{\partial t}}\left(f_k\right)F^{\prime }\left(f_k\right)\phi ,\hfill \end{array}$$

we need to prove that

$$\begin{array}{cc}& {\displaystyle {\int }_{\delta }^T\int F^{\prime }\left(f_k\right){\langle \nabla f_k,\nabla \phi \rangle }_g+{\int }_{\delta }^T\int \frac{\partial }{\partial t}\left(f_k\right)F^{\prime }\left(f_k\right)\phi }\hfill \\ & =-{\int }_{\delta }^T\int -{\Delta }_{\mu }F\left(f_k\right)\phi -{\int }_{\delta }^T\int {\left|{\nabla }_g{f}_k\right|}_g^2\phi F\left(f_k\right)-{\int }_{\delta }^T\int {\partial }_t{f}_k\phi F\left(f_k\right)\hfill \\ & =-{\int }_{\delta }^T\int {\langle \nabla f_k,\nabla \phi \rangle }_{g}F\left(f_k\right)d\mu dt-{\int }_{\delta }^T\int {\left|{\nabla }_g{f}_k\right|}_g^2\phi F\left(f_k\right)-{\int }_{\delta }^T\int {\partial }_t{f}_k\phi F\left(f_k\right).\hfill \end{array}$$

Since

$$F^{\prime }\left(f_k\right){\langle \nabla f_k,\nabla \phi \rangle }_g=-e^{-f_k}{\langle \nabla f_k,\nabla \phi \rangle }_g,$$

and

$$\begin{array}{cc}\hfill {\displaystyle -{\langle \nabla f_k,\nabla \phi \rangle }_{g}F\left(f_k\right)}& -|{\nabla }_g{f}_k{|}_g^2\phi F\left(f_k\right)\hfill \\ & =-\phi {\langle \nabla f_k,\nabla F\left(f_k\right)\rangle }_g-F\left(f_k\right){\langle \nabla f_k,\nabla \phi \rangle }_g-{\left|{\nabla }_g{f}_k\right|}_g^2\phi F\left(f_k\right)\hfill \\ & =-\phi F^{\prime }\left(f_k\right)|{\nabla }_g{f}_k{|}_g^2-F\left(f_k\right){\langle \nabla f_k,\nabla \phi \rangle }_g-{\left|{\nabla }_g{f}_k\right|}_g^2\phi F\left(f_k\right)\hfill \\ & =\phi e^{-f_k}|{\nabla }_g{f}_k{|}_g^2-e^{-f_k}{\langle \nabla f_k,\nabla \phi \rangle }_g-{\left|{\nabla }_g{f}_k\right|}_g^2\phi e^{-f_k}\hfill \\ & =\phi e^{-f_k}|{\nabla }_g{f}_k{|}_g^2-{\left|{\nabla }_g{f}_k\right|}_g^2\phi e^{-f_k}-e^{-f_k}{\langle \nabla f_k,\nabla \phi \rangle }_g,\hfill \end{array}$$

for all test function $\phi$, we have

$$F^{\prime }\left(f_k\right){\langle \nabla f_k,\nabla \phi \rangle }_g=-{\langle \nabla f_k,\nabla \phi F\left(f_k\right)\rangle }_g-{\left|{\nabla }_g{f}_k\right|}_g^2\phi F\left(f_k\right).$$

Hence, $V_k\to V$ in $L^2\left([\delta ,T]\times M,d\mu dt\right)$ and $f_k\to f,$$\mu -a.e$.

It follows, from Definition 1, that $V={\Delta }_{\mu }f-{\left|{\nabla }_{g}f\right|}_g^2-{\displaystyle \frac{\partial }{\partial t}}f.$ So, using (16), we obtain $u:=e^{-f}\in L^2([\delta ,T]\times M,d\mu dt).$

As an example of a function satisfying (6), where the existence part in the classical result [11] is recovered, we can take the following form.

Example 1

([11]). For $H=-\Delta$, $M:=B(0,r_0)\subset {\mathbb{R}}^d$, $|{\nabla }_g{f|}_g=|\nabla f|$ and $d\ge 3$. Let $\alpha \in \mathbb{R}$ and $V\left(x\right)={\displaystyle \frac{\alpha (d-2-\alpha )}{{\left|x\right|}^2}}$, then

$$V=\Delta f-{|\nabla f|}^2-{\displaystyle \frac{\partial f}{\partial t}}+2{\displaystyle \frac{\nabla \phi }{\phi }}.\nabla f,$$

where φ be the normalized positive eigenfunction corresponding to the first eigenvalue of $-\Delta$ on B and $f=ln\left(r^{\alpha }\phi \left(r\right)e^{-\lambda t}\right)$ with $r=\left|x\right|$, which implies that (1) has one positive solution.

Our second result in this paper is the following.

Theorem 2.

Assume that $V={\Delta }_{\mu }f-{\left|{\nabla }_{g}f\right|}_g^2-{\displaystyle \frac{\partial }{\partial t}}f,$ for some f such that $e^{-f}\in L^2(M\times [\delta ,T],d\mu dt)$ and $e^{-f(.,t)}\in W^1\left(M\right)$. Then, for any given non-negative function $u_{\delta }$ in $L^2(M,\mu )$, (1) has a non-negative solution $u\in L^2(M\times [\delta ,T],d\mu dt)$ such that $u(.,t)\in W^1\left(M\right)$.

Proof. 

Let f be such that $e^{-f}\in L^2(M\times [\delta ,T],d\mu dt)$ and $e^{-f(.,t)}\in W^1\left(M\right)$. Put $V={\Delta }_{\mu }f-{\left|{\nabla }_{g}f\right|}_g^2-{\displaystyle \frac{\partial }{\partial t}}f.$ Then, according to our assumption, there exist sequences of functions $\left\{V_k\right\}$ and $\left\{f_k\right\}$ satisfying

$$\parallel V_k{\parallel }_{L^2}\le C,|V_k|\le |V_{n+1}|,f_k\in L^2\left([\delta ,T]\times M,d\mu dt\right),{\parallel V_k-V\parallel }_{L^2}\to 0$$

and

$$V_k={\Delta }_{\mu }{f}_k-{\left|{\nabla }_g{f}_k\right|}_g^2-{\displaystyle \frac{\partial }{\partial t}}{f}_k.$$

Input $u_k(x,t)=e^{-f_k(x,t)}$ for all $(x,t)\in M\times (\delta ,+\infty )$. Let $F\left(u_k\right)=e^{-f_k}$, which gives ${\displaystyle \frac{\partial }{\partial t}}\left[F\left(u_k\right)\right]=-f_k^{\prime }{e}^{-f_k}$. Then, for all test function $\phi$, we have

$$\begin{array}{cc}\hfill {\displaystyle {\int }_{\delta }^t\int -{\Delta }_{\mu }{u}_k\phi }& +{\int }_{\delta }^t\int {\partial }_t{u}_k\phi -{\int }_{\delta }^t\int V_k{u}_k\phi \hfill \\ \hfill =& {\int }_{\delta }^t\int -{\Delta }_{\mu }F\left(f_k\right)\phi +{\int }_{\delta }^t\int {\partial }_t{f}_k{F}^{\prime }\left(f_k\right)\phi -{\int }_{\delta }^t\int V_{k}F\left(f_k\right)\phi \hfill \\ \hfill =& {\int }_{\delta }^t\int {\langle \nabla F\left(f_k\right),\nabla \phi \rangle }_g+{\int }_{\delta }^t\int {\partial }_t{f}_k{F}^{\prime }\left(f_k\right)\phi \hfill \\ & -{\int }_{\delta }^t\int \left[{\Delta }_{\mu }{f}_k-{|\nabla f_k|}_g^2-{\displaystyle \frac{\partial }{\partial t}}{f}_k\right]F\left(f_k\right)\phi \hfill \\ \hfill =& {\int }_{\delta }^t\int {\langle \nabla F\left(f_k\right),\nabla \phi \rangle }_g+{\int }_{\delta }^t\int {\partial }_t{f}_k{F}^{\prime }\left(f_k\right)\phi +{\int }_{\delta }^t\int -{\Delta }_{\mu }{f}_k\phi F\left(f_k\right)\hfill \\ & +{\int }_{\delta }^t\int {|\nabla f_k|}_g^{2}F\left(f_k\right)\phi +{\int }_{\delta }^t\int {\partial }_t{f}_k\phi F\left(f_k\right)\hfill \\ \hfill =& {\int }_{\delta }^t\int F^{\prime }\left(f_k\right){\langle \nabla f_k,\nabla \phi \rangle }_g+{\int }_{\delta }^t\int {\langle \nabla f_k,\nabla \phi \rangle }_{g}F\left(f_k\right)\hfill \\ & +{\int }_{\delta }^t\int {|\nabla f_k|}_g^{2}F\left(f_k\right)\phi -{\int }_{\delta }^t\int {\partial }_t{f}_{k}F\left(f_k\right)\phi +{\int }_{\delta }^t\int {\partial }_t{f}_k\phi F\left(f_k\right).\hfill \end{array}$$

Note that

$$\begin{array}{cc}\hfill {\displaystyle F^{\prime }\left(f_k\right)\nabla f_k\nabla \phi }& +\nabla f_k\nabla \phi F\left(f_k\right)+{|\nabla f_k|}_g^{2}F\left(f_k\right)\phi \hfill \\ & =F^{\prime }\left(f_k\right){\langle \nabla f_k,\nabla \phi \rangle }_g+\phi {\langle \nabla f_k,\nabla F\left(f_k\right)\rangle }_g+F\left(f_k\right){\langle \nabla f_k,\nabla \phi \rangle }_g+{|\nabla f_k|}_g^{2}F\left(f_k\right)\phi \hfill \\ & =F^{\prime }\left(f_k\right){\langle \nabla f_k,\nabla \phi \rangle }_g+\phi F^{\prime }\left(u_k\right)|\nabla f_k{|}_g^2+F\left(u_k\right){\langle \nabla f_k,\nabla \phi \rangle }_g+{|\nabla f_k|}_g^{2}F\left(f_k\right)\phi \hfill \\ & =-e^{-f_k}{\langle \nabla f_k,\nabla \phi \rangle }_g-\phi e^{-f_k}|\nabla f_k{|}_g^2+e^{-f_k}{\langle \nabla f_k,\nabla \phi \rangle }_g+{|\nabla f_k|}_g^2\phi e^{-f_k}=0.\hfill \end{array}$$

Therefore, we obtain ${\Delta }_{\mu }{u}_k+V_k{u}_k-{\displaystyle \frac{\partial u_k}{\partial t}}=0,\mu -a.e.$ In what follows, we show that the sequence $\parallel u_k{\parallel }_{L^2}$ is uniformly bounded. From Definition 1, $f_k(x,\delta )$ is independent of k, and thus, $u_k(x,\delta )=u_{k+1}(x,\delta )\mathrm{for}\mathrm{all}x\in M.$

By the maximum principle [19], we have $0\le u_k\le u_{k+1}.$

Now using the assumption that $f_k\to f$ as $k\to \infty$, we deduce that

$$u_k\to e^{-f}ask\to \infty .$$

(17)

Note that $e^{-f}\in L^2(M\times [\delta ,T],d\mu dt)$ and $e^{-f(.,t)}\in W^1\left(M\right)$. From (17), for every $ϵ>0$, we obtain $\parallel u_k{\parallel }_{L^2}\le \parallel u_k{-u\parallel }_{L^2}+{\parallel u\parallel }_{L^2}\le ϵ+{\parallel u\parallel }_{L^2}.$ So, $\parallel u_k{\parallel }_{L^2}$ is uniformly bounded. With the weak compactness in $L^2(M\times [\delta ,T],d\mu dt)$, there exists a subsequence, still denoted by $\left\{u_k\right\}$, such that $u_k$ converges weakly to u in $L^2(M\times [\delta ,T],d\mu dt)$. Let $\varphi$ be a test function; then, we have

$$\begin{array}{cc}\hfill \parallel u_k{V}_k{\varphi -uV\varphi \parallel }_{L_{loc}^1}& =\parallel u_k{V}_k\varphi -u_{k}V\varphi +u_k{V\varphi -uV\varphi \parallel }_{L_{loc}^1}\hfill \\ & \le \parallel u_k(V_k-V){\varphi \parallel }_{L_{loc}^1}+{\parallel (u_k-u)V\varphi \parallel }_{L_{loc}^1}\hfill \\ & \le \parallel u_k{\parallel }_{L^2}.\parallel V_k{-V\parallel }_{L^2}{.\parallel \varphi \parallel }_{L^{\infty }}+\parallel (u_k-u){V\parallel }_{L_{loc}^1}.{\parallel \varphi \parallel }_{L^{\infty }}.\hfill \end{array}$$

Therefore, we have $\parallel u_k{V}_k{\varphi -uV\varphi \parallel }_{L_{loc}^1}\to 0asn\to \infty .$ Consequently, u is a weak solution for the problem posed in (1). Moreover, since, for all integers k, $u_k\ge 0$, u is a non-negative solution for the problem posed in (1). This completes the proof of Theorem 2. □

3. Conclusions

In this paper, we studied a parabolic problem involving the Laplace–Beltrami operator on a smooth compact manifold. This work introduces a novel class of time-dependent potentials expressed as a combination of function derivatives; moreover, we presented a new approach to discuss the existence of a fundamental solution to the heat equation on a Riemannian manifold.

Based on the same idea, we will study some elliptic equations with potential as spatial derivatives of some functions. Moreover, we will study the blow-up and the regularity of such solutions.