On Non-existence of Global Weak-predictable-random-field Solutions to a Class of SHEs

The multiplicative non-linearity term is usually assumed to be globally Lipschitz in most results on SPDEs. This work proves that the solutions fail to exist if the non-linearity term grows faster than linear growth. The global non-existence of the solution occurs for some non-linear conditions on $\sigma$ . Some precise conditions for existence and uniqueness of the solutions were stated and we have established that the solutions grow in time at most a precise exponential rate at some time interval; and if the solutions satisfy some non-linear conditions then they cease to exist at some finite time t . Our result also compares the non-existence of global solutions for both the compensated and non-compensated Poisson noise equations


Introduction
Stochastic PDEs have now received much interest, see [1,2,3] and their references; but the blow up result or rather the non-existence of global result for the discontinuous or jump process has not been given. The study of blow up or global non-existence of solution is as old as Mathematics itself, see [4,5]. The clarity between blow up of a solution and non-existence of global solution of a deterministic PDEs was given by Sugitani, see [6] and the references. Our mild solution here, is a weak-predictable random field solution since it is a class of stochastic heat equation with jump-type process (poisson random noise measure), see [7,8]. We therefore show conditions where the second moment of the solution to the compensated equation and the first moment of the solution to the non-compensated equation (otherwise known as energy solutions) fail to exist and compare the non-existence of global solutions of the two equations with each other. We also consider /2 ) ( = : , the generator of α -stable processes and use some explicit bounds on its corresponding fractional heat kernel to obtain more precise results. We also show that when the solutions satisfy some non-linear growth conditions on σ , the solutions cease to exist for both compensated and noncompensated noise terms for different conditions on the initial function ) ( 0 x u . Now, consider the following stochastic heat equations driven by a compensated and non-compensated Poisson noises (discontinuous processes). We follow the method of [9] in making sense of our integral solutions.
where ,.,.) (t p is the heat kernel. If in addition to the above,

Definition 1.2 ( random field solution)
We seek a mild solution to equation (1.2) of the form.
with ,.,.) (t p the heat kernel. We impose the following integrability condition on the solution: We now make sense of the discontinuous integrals by stating the existence theorem (Ikeda and Watanabe [11]) characterised by a stationary Point process.
Proof. Ikeda and Watanabe [11]. Now applying the above theorem with One set of the vectors will play the role of position while the other will play the role of "jumps". By the above theorem, we have a Poisson point process and N denotes the null set of F . We can write the Poisson random measure as )), ( ), In this case, we have . We now describe the stochastic integral with respect to this Poisson random measure and define the class of integrand precisely: The following integral can now be defined for all as the a.s sum of the following absolutely convergent sum.
as the a.s sum of the following absolutely convergent sum.
For an example, one can define a Poisson random measure with the following property: Therefore the above integral is finite since the sum contains finitely many terms.
For the existence and uniqueness of (1.1), we need the following condition on σ . Essentially this condition says that σ is globally Lipschitz in the first variable and bounded by another function in the second variable.

Condition 1.6 There exist a positive function J and a finite positive constant,
The function J is assumed to satisfy the following integrability condition: where K is some finite positive constant.
For the lower bound result, we will need the following extra condition on σ .

Condition 1.7 There exist a positive function J and a finite positive constant, σ L such that for all
The function J is assumed to satisfy the following integrability condition: where K is the constant from (1.9) and κ is another positive, finite constant.
For the existence and uniqueness of (1.2), we make the following assumption.

Condition 1.8 There exist a positive function J and a finite positive constant,
The function J is assumed to satisfy the following integrability condition.
where K is some finite positive constant.
We can also give the lower bound estimate on the growth of the first moment.

Condition 1.9
There exist a positive function J and a finite positive constant, σ L such that for all The function J is assumed to satisfy the following integrability condition.
where K is the constant from (1.13) and κ is another positive finite constant.
The outline of the paper is given below. The paper is comprised of five sections. Statements and conditions of main results are given in section two. In section three, some estimates on heat kernels are given, some known results and few new propositions are also given. We also give estimates where the proofs of our theorems lie. The proofs of our main results are given in section four and section five contains a brief conclusion of the research.

Main Results
We show that if the function σ grows faster than linear growth, then the energy of the solutions, that is, the second moment Suppose that instead of (1.10), we have the following condition.

Condition 2.1 There exists a constant
where the constant σ L and the function J are the same as in condition 1.7.
We then have the following result. If instead of (1.14), we consider the following condition: where the constant σ L and the function J are the same as in condition 1.9.
We then have the following result with the initial condition where µ is a Lebesgue measure.

Some auxiliary results and estimates
We present some properties of ) , ( x t p that will be used in the proof of our results, see [6].
From the above relation, , is a decreasing function of t . The heat kernel ) , ( x t p is also a decreasing function of We now state some propositions and lemma whose proofs can be found in [12].
Proposition 2.5 [12]. Let ) , ( x t p be the transition density of a strictly α -stable process. If [13,14,15,16] With the assumption that the initial condition 0 u is positive on a set of positive measure, we then have the following.
Proposition 2.7 [12]. There exist a 0 > T and a positive constant 1 c such that for all T t > and all Proof. The proof follows by applying Lemma 2.6.
The next proposition is similar to but more general than Proposition 2.7.
Proposition 2.8 [12]. Given the assumption on the initial function 0 Here also, we present the time continuity of the first moment of the solution (1.2). Proposition 2.10 [12]. Suppose that condition 1. 8 We now give the following proposition which establishes the fact that under the local Lipschitz continuity as stated in condition 2.1, there exists a unique solution up to a fixed time T . [12].

Proposition 2.11
( h x N σ therefore satisfies (1.8) but with a different constant. The result follows by the proof of existence and uniqueness of the solution and its time continuity above.
We will also need the following proposition which establishes the fact that under the local Lipschitz continuity as stated in condition 2.2, there exists a unique solution up to a fixed time T . Proposition 2.12 [12]. Suppose that condition 2. Here follows Jensen's inequality which shall be used in the proof of the blow-up result. [12]

Lemma 2.13
The above inequality reduces to We now use Lemma 2.6 to find lower bounds on the heat kernel appearing in the above display.
(t Y ceases to exist in finite time for all β such that α β < < 1 We now prove the finite time blow up for the non-compensated noise equation.
Proof of Theorem 2.4. We begin with the first part of the theorem. Now write We use the assumption that We now solve the following differential inequality: (0) c F ≥ or simply the ordinary differential equation: We perform a change of variable, by letting , and integrating in x d , we obtain