 Thank you for giving me the chance to talk today. So my work is entitled A Stochastic Mass-Conserved Reaction Diffusion Equation with Non-Linear Diffusion. This work is a joint work with Kaili, a PhD student at the University of Tokyo in Japan, and with my PhD advisor Daniel Hillhorst, which is in the University of Paris-Duit. So first to start, a particular or an example of the reaction diffusion equation is the mass-conserved Allen-Kahn equation. So we introduce the problem partial derivative of phi over partial derivative of t equal to Laplacian phi, which is a linear diffusion, plus f phi, which is a reaction term, minus 1 over the measure of the integral of the f phi, which is the non-local term, special Neumann boundary conditions, and an initial condition, which is phi 0. f, the reaction term here, has exactly three zeros, minus 1, 0, and plus 1. f prime on plus and minus 1 is negative, and f prime in zero is positive. A typical example of this reaction term is f phi equals 2 phi minus phi cube. This deterministic equation with linear diffusion and with the deterministic one, the mass-conserved Allen-Kahn, is, it was introduced by Rubinstein and Steinberg in 1992, and it is two model binary mixtures undergoing phase separation. Then now we introduce the stochastic mass-conserved Allen-Kahn equation. So it's the same equation with a linear diffusion, the reaction term, the non-local term, and we add the noise term, the stochastic term, which is the partial derivative of w over the partial derivative of t, Neumann boundary conditions, and phi 0, an initial condition. The singular limit of this equation was studied in the article of Antonopoulos, Bates, Blomker, and Karali in 2016, and this problem was to model the motion of a droplet. However, they left the question of existence and uniqueness of the solution of the problem open. So our goal is to study the non-local stochastic reaction diffusion equation with nonlinear diffusion. So I did introduce the following problem. So it's the partial derivative of phi over partial derivative of t, divergence equal to divergence of a of gradient phi, the nonlinear diffusion term, plus f phi, which is the reaction term and which is more general in the case that I studied, but here I will consider the one of Alenkaard for simplicity, and then the non-local term plus the stochastic term, some special boundary conditions, and an initial condition phi 0. D, in my case, it's an open bounded set of Rn with sufficiently smooth boundary. So my goal is to study the existence and uniqueness of the problem p, of the solution of the problem p. A is Lipschitz continuous from Rn to Rn, and A is coercive, so we'll put some coercivity condition. This, the divergence of A of gradient of u, this nonlinear diffusion term, was introduced in an article of Funakian's Spoon in 1997. We remarked that if A equals to the identity matrix, then the divergence of A of gradient of u equals to Laplacian of u. Then, now the function w, it's a cube, a cube Brownian motion on L2. So w, it's equal to the infinite sum from k equals to 1 to infinity of the square root of lambda k, beta k, e k. E k are the orthonormal basis on L2, diagonalizing q, lambda k are the corresponding eigenvalues, and q is a nonnegative definite symmetric operator on L2 with the trace, with a finite trace. The trace of q, we suppose moreover that the trace of q, which is the infinite sum of the lambda k, are bounded by a constant big lambda zero, which is positive. And finally, the beta k is a sequence of independent Brownian motion on some probabilistic play, on some probabilistic domain. Then now to start the proof of existence, we have to introduce a preliminary, we have to introduce an auxiliary problem. So we take the equation of the heat equation, the stochastic heat equation, with nonlinear diffusion term. So I introduce namely the problem P1, which is partial derivative of WA over partial derivative of T, equals to divergence of A of gradient of WA plus the stochastic term. Some boundary conditions and initial condition equals to zero. This problem was the well-positiveness of this problem, the existence and the uniqueness of the solution, were studied in an article of Krilov and Drosovsky in 2007. So now we give the definition of this problem. WA belongs to some sub-left spaces, and WA satisfies the integral equation, which is WA equals to the integral from zero to T of divergence of A of gradient of WA plus WA of x and t. Then I define a change of function. I say that U is equal to phi minus WA. Then U satisfies the problem P2, which is partial derivative of U over partial derivative of T, equal to the nonlinear diffusion term, which is divergence of A of gradient of U plus WA minus A gradient WA. The reaction term, F of U plus WA minus the nonlocal term, one over measure of the integral of U plus WA. And very marked here that the noise is hidden. Then we have the special boundary condition and phi zero, the initial condition. We remarked that this equation is mass conserved. So, namely, we have that the integral over D of U equals to the integral over D of phi zero, almost surely for almost every T belonging to R plus. Then we work on some sub-left spaces, which are particular to have the existence of the solution. So we introduce H, which is V belonging to L2 of D, the sub-left space, such that the integral over D of V equals to zero. Then I introduce capital V, big V equals to H1 intersection with H, and then Z equals to V intersection with L2P. Then I define again the solution now of problem P2. U belongs to some sub-left space, and then U satisfies the integral equation U of X and T equals to phi zero plus integral from zero to T of the nonlinear diffusion term plus integral over zero to the reaction term minus the nonlocal term. Now I will prove the theorem there exists a unique solution of the problem P2. For the proof, I will give some ideas. We apply a Galerkin method. The Galerkin method is used to transform a continuous problem to a discrete problem, and then the solution of the Galerkin approximate problem can be decomposed on a basis of functions. So I introduce gamma 1, gamma 2, gamma k, the eigenvalues of the operator minus Laplacian with homogeneous Neumann boundary conditions, and then Wk, which are the eigenfunctions on L2, and they are the eigenfunctions of the operator minus Laplacian. Wk are smoothed because we have that the boundary of our domain is sufficiently smoothed. Then now I prove the following lemma. The function Wj are an orthonormal basis of L2, and the first eigenfunction W0 is equal to a constant, namely one over the square root of the measure of D. To prove that they are orthonormal basis, this is very easy. So we see that the integral over D, we want to prove that this is equal to zero, but these are the eigenfunction of minus Laplacian. So it's namely equal to one minus one over a gamma j integral over D of Laplacian Wj. But we have Neumann boundary conditions, then this one will be equals to zero for all j different than zero. And this is how we prove the lemma. Then now, as I said, the solution of the Gallerkin problem. So we look for an approximate solution um minus big m, which is equal to the sum from i equal one to m of uim w i. Capital m is equal to one over the measure of D, integral over D of phi zero, the initial condition. This solution satisfies the following equation, integral over D of partial derivative of um minus capital m over partial derivative of t multiplied by Wj minus integral over D. Here, this is the nonlinear diffusion term, which is integrated by part multiplied by the gradient of Wj. Then we add the integral over D of f um plus WA multiplied by Wj, Wj, and then the nonlocal term multiplied by Wj. We have also that um for t equals zero converges strongly in L2 to phi zero, the initial condition as m goes to infinity. We remarked that the contribution of the nonlocal term vanishes. And this is because the integral over D of Wj, as I said here, is equal to zero. So the nonlocal term vanishes. And then um minus m verifies a simple, a more simple equation. Then we proved that we have some a priori estimates, so there exists a positive constant, which bound the solution in some sub-f spaces. And then from this, from these a priori estimates, I can say that there exists a subsequence such that um minus capital m converges weakly into some sub-f space. And there exists chi and phi such that the reaction term converges weakly to chi in some sub-f space. And the nonlinear diffusion term converges weakly to phi in some sub-f space. Then we pass to the limit as m goes to infinity from the from the equation of um minus capital m to the limit equation. So we get this equation here, but in the nonlinear diffusion term we have the limit, which is phi minus divergence of A of gradient of W. And here we have the limit of the reaction term, which is chi. The last point of the proof is to see how, to identify the limit. So to have that phi plus chi equals to divergence of A of gradient of U plus W A plus F of U plus W A. And to do this, we have to use a monotonicity argument. This monotonicity argument was introduced by in an article by Martin Marion in 1987 for deterministic problem and for stochastic problems in an article of Krilov and Rolazovsky in 2007. So to apply the monotonicity argument, first for the nonlinear diffusion term, we have to use the condition of A, the condition of coercivity of A that I said in the first slide to deduce that this term is monotone. Then for the reaction term, I would take the simplest one, which is Fu equals U minus U cube. We have to see that this one, the U cube, is a function which is monotone. And we have to get rid of this one, which is not monotone. So we do some change of function, which is exponential to the power Ct U tilde with C equals one in particular in this case of U minus U cube. And this is how we get that this one is monotone. And finally, for the nonlocal term, we have that one over the measure over D of Fu multiplied by the integral of U. But we are working on some space, some sub-life space, such as the integral of D of the solution equals to zero. Then this term also vanishes. Last thing I wanted to do is to prove the uniqueness to do this. I take two solutions, U1 minus U2. So those two solutions verifies the equation U1 minus U2 equals to the nonlinear diffusion term of the first one minus the second, the reaction term of the first minus the second, the same for the nonlocal term. And then we take its duality product with U1 minus U2. We have the same initial conditions for U1 and U2. They are equal to phi zero. So the contribution of the nonlocal term is also equal to zero. And then we take the expectation of the equation. And by granular lemma, we prove that U1 equals U2 for almost every omega x and t belonging to big omega cross D cross zero t. Thank you for your attention. What is the parameter p, the power p that appeared in, yeah, here? Okay. Actually, I did not mention this because, okay. So I said that fU is equal to U minus U cubed and the case of Allen-Cahn. But in our case, what we did is that we extended this one to a more general polynomial. So we have that U is equal to the sum of some coefficient G from 1 to 2p minus 1. So it's a more, it's a more, how to say, it's a more general polynomial such that p is bigger than 2. So we did not consider only the combination U minus U cubed, but you can have more polynomial to plus U to p minus 1. Is there any other question? So compared to the solution of the equation, data communist, the solution of the equation stochastic, the compartment of the two has some difference? I do not know. I did not do numeric simulations. So I don't know the difference between the one in deterministic and the one in stochastic. But what can I say is that I have the existence of uniqueness of the first in deterministic and I proved the one in stochastic. But to compare the solution, I don't know because I did not do any numeric simulations. And I have a last question. What kind of physical mechanism can give a nonlinear diffusion term? And for a linear diffusion term, it's just, okay, buoyant motion or random works. But what kind of mechanism? For the A, not to be the identity. Okay. So I can tell you what kind of this diffusion physically can appear. So if you take the normal diffusion, if we take, I don't know, the tumor or something biological and we want to see the diffusion, in the Laplacian case, you can see that this one can diffuse with the same velocity. But in the case where we put this one, you can imagine that you can have different velocities. So this one is here, the gradient. So you can have the component of x that goes with the velocity different than the one of the y. So you can have another shape and not a circular shape diffusion. So I don't know, it's something like this and going with really different velocities. Okay. Thank you.