**INTRODUCTION**

We consider the problem

This equation models many physical processes, which include time-dependent, irreversible processes such as that of conduction, chemical reactions and biological flow problems. Thus many authors in various forms have investigated the problem imposing different boundary conditions which suit the situations at hand and using various forms of f (x, t, u). The type of solution one gets depends on the form of f(x, t, u) and the boundary conditions. The solution u(x,t) can be defined for all positive t, in which case we call it a global solution or unbounded in finite time, in which case we say it blows up.

Escobedo and Herrero^{[1]} and Fujita and Watanabe^{[2]} considered

where, Ω is a bounded open subset of R^{N} and established the existence and uniqueness of positive solution. Assuming that the initial value u_{0} is non-negative, they found the solution as:

for all

Rossi^{[3]} obtained the blow up rate for positive solution of:

with boundary conditions

In this case one has a non-linear term at the boundary and a reaction term in the equation. If λ>0, these 2 terms compete and blow up phenomenon occurs if p<2q-1 or p = 2q-1.

They established that if p<2q-1 or p = 2q-1 the blow up rate is given by:

When λ>0, Chipot *et al*.^{[4]} and Lopez *et al*.^{[5]}
proved the existence and regularity of solution for initial data that satisfies
compatibility condition. They found out that the solution of eqn. (3) only exists
for a finite period of time. Cazenave *et al*.^{[6]} introduced
the concave and convex term in the equation in the form:

including a Dirichlet boundary condition. The non-linearity on the RHS is the
sum of the concave and convex term with the non-linearity being singular at
0 (it is not Lipschitz because q<1). They showed that there exists a global
solution if and only if there exists a weak solution of the stationary equation.
Then Haraux^{[7]} considered a linear parabolic equation with Lipschitz
continuous boundary condition

and showed that there is a unique global solution. Using this he established
that for

then

In a recent work, Messaoudi^{[8]} proved the local existence for the
problem

and showed that the solution blows up in finite time.

In the realm of the classical theory of differential equations, it can be shown
imposing a Lipschitz condition on the non-linearity term f that the equation
has a unique solution if u_{0}(x) ∈ C^{∞} (Ω).
On the other hand if u_{0}(x) is a distribution with compact support,
the system has no solution in the classical sense^{[9]}. Thus Ifidon
and Oghre^{[10]} formulated a generalized function G(Q), using classical
estimates and induction hypothesis over the order of the differential operators
which defines the element of G(Q) to prove the existence, uniqueness as well
as consistence results for the solution to the problem.

This study considers:

and use an inverse process of Fourier transform to determine the solution u(x, t).

**METHOD OF SOLUTION**

Let u(x, t) be the solution and assume that both u(x, t) and f(x, t) are piecewise smooth and integrable over (0, ∞), then by Fourier cosine transform

Differentiating (11) with respect to t and substituting in (10)

where, F_{c} is the Fourier cosine transform of f(x, t).

Evaluating the integral on the right using integration by parts

This is a linear differential equation whose integrating factor is given as

Taking the inverse Fourier cosine transform

Interchanging the order of integration, the first integral becomes

Using the fact that being odd then

Then equation (18) becomes

Also the second integral in (16) becomes

Hence by (19) and (21)

**CONCLUSION**

We have considered the one dimensional semi-linear heat equation and we have considered a piecewise continuous integrable function f(x, t). Using a Fourier cosine transform and a backward inverse operation we have determined the solution to the problem.